Search: a014613 -id:a014613
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A101638
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Number of distinct 4-almost primes A014613 which are factors of n.
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+20
15
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1
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OFFSET
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1,48
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COMMENTS
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This is the inverse Moebius transform of A101637. If we take the prime factorization of n = (p1^e1)*(p2^e2)* ... * (pj^ej) then a(n) = |{k: ek>=4}| + ((j-1)/2)*|{k: ek>=3}| + C(|{k: ek>=2}|,2) + C(j,4). The first term is the number of distinct 4th powers of primes in the factors of n (the first way of finding a 4-almost prime). The second term is the number of distinct cubes of primes, each of which can be multiplied by any of the other distinct primes, halved to avoid double-counts (the second way of finding a 4-almost prime). The third term is the number of distinct pairs of squares of primes in the factors of n (the third way of finding a 4-almost prime). The 4th term is the number of distinct products of 4 distinct primes, which is the number of combinations of j primes in the factors of n taken 4 at a time, A000332(j), (the 4th way of finding a 4-almost prime).
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REFERENCES
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Hardy, G. H. and Wright, E. M. Section 17.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.
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LINKS
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M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
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EXAMPLE
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a(96) = 2 because 96 = 16 * 6 hence divisible by the 4-almost prime 16 and also 96 = 24 * 4 hence divisible by the 4-almost prime 24.
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PROG
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(PARI) a(n)=my(f=factor(n)[, 2], v=apply(k->sum(i=1, #f, f[i]>k), [0..3])); v[4] + v[3]*(v[1]-1) + binomial(v[2], 2) + v[2]*binomial(v[1]-1, 2) + binomial(v[1], 4) \\ Charles R Greathouse IV, Sep 14 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A109024
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Numbers that have exactly four prime factors counted with multiplicity (A014613) whose digit reversal is different and also has 4 prime factors (with multiplicity).
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+20
11
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126, 225, 294, 315, 459, 488, 492, 513, 522, 558, 621, 650, 738, 837, 855, 884, 954, 1035, 1062, 1098, 1107, 1197, 1206, 1236, 1287, 1305, 1422, 1518, 1617, 1665, 1917, 1926, 1956, 1962, 1989, 2004, 2034, 2046, 2068, 2104, 2148, 2170, 2180, 2223, 2226
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OFFSET
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1,1
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COMMENTS
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This sequence is the k = 4 instance of the series which begins with k = 1, k = 2, k = 3 (A109023).
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LINKS
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Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.
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EXAMPLE
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a(1) = 126 is in this sequence because 126 = 2 * 3^2 * 7 is a 4-almost prime and reverse(126) = 621 = 3^3 * 23 is also a 4-almost prime.
a(2) = 225 is in this sequence because 225 = 3^2 * 5^2 is a 4-almost prime and reverse(225) = 522 = 2 * 3^2 * 29 is also a 4-almost prime. That 225 and 522 are concatenated from entirely prime digits is a coincidence, as with 2223).
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MATHEMATICA
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Select[Range[2226], PrimeOmega[#]==4 && PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==4 &&!PalindromeQ[#]&] (* James C. McMahon, Mar 07 2024 *)
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PROG
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(PARI) is(n) = {
my(r = fromdigits(Vecrev(digits(n))));
n!=r && bigomega(n) == 4 && bigomega(r) == 4
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CROSSREFS
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Cf. A006567, A097393, A109018, A109023, A109025, A109026, A109027, A109028, A109029, A109030, A109031.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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29, 53, 97, 113, 161, 159, 145, 269, 244, 232, 231, 247, 261, 373, 399, 386, 328, 350, 375, 371, 395, 547, 559, 572, 537, 541, 577, 635, 679, 663, 607, 621, 687, 673, 658, 769, 871, 853, 839, 856, 832, 881, 947, 939, 1003, 1007, 955, 915, 907, 889, 941, 989
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OFFSET
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1,1
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LINKS
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FORMULA
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EXAMPLE
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a(1) = prime(4almostprime(1)) - 4almostprime(prime(1)) = 53 - 24 = 29.
a(2) = prime(4almostprime(2)) - 4almostprime(prime(2)) = 89 - 36 = 53.
a(3) = prime(4almostprime(3)) - 4almostprime(prime(3)) = 151 - 54 = 97.
It is mere coincidence that the first 4 values are all primes.
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MATHEMATICA
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FourAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@ Sqrt[n/(Prime@i*Prime@j)]}];
FourAlmostPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[ FourAlmostPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2];
Table[ Prime@ FourAlmostPrime@ n - FourAlmostPrime@ Prime@ n, {n, 52}]
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PROG
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(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot, prime
def f(x): return int(x-sum(primepi(x//(k*m*r))-c for a, k in enumerate(primerange(integer_nthroot(x, 4)[0]+1)) for b, m in enumerate(primerange(k, integer_nthroot(x//k, 3)[0]+1), a) for c, r in enumerate(primerange(m, isqrt(x//(k*m))+1), b)))
m, k = n, f(n)+n
while m != k:
m, k = k, f(k)+n
r, k = (p:=prime(n)), f(p)+p
while r != k:
r, k = k, f(k)+p
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CROSSREFS
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Cf. Primes indexed by 4-almost primes = A124282. 4-almost primes indexed by primes = A124283. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)).
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KEYWORD
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easy,nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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16, 384, 13824, 552960, 29859840, 1672151040, 100329062400, 8126654054400, 682638940569600, 60072226770124800, 5406500409311232000, 540650040931123200000, 56227604256836812800000
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OFFSET
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1,1
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COMMENTS
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4-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086046 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008586(n) = 4*n.
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LINKS
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FORMULA
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a(n) = Prod[from i = 1 to n] A014613(i).
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EXAMPLE
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a(5) = 29859840 = 16 * 24 * 36 * 40 * 54 = the product of the first 5 values of the 4-almost primes = 2^13 * 3^6 * 5, which has 4*5 = 20 prime factors (with multiplicity).
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MATHEMATICA
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FoldList[Times, Select[Range[200], PrimeOmega[#]==4&]] (* Harvey P. Dale, Dec 02 2018 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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8, 12, 4, 14, 2, 4, 21, 3, 4, 2, 10, 4, 22, 6, 3, 1, 4, 10, 2, 4, 28, 5, 7, 2, 6, 6, 10, 5, 3, 4, 2, 14, 2, 10, 16, 18, 2, 1, 9, 2, 7, 13, 2, 10, 2, 2, 4, 2, 1, 13, 8, 3, 1, 4, 10, 24, 10, 17, 3, 15, 1, 2, 10, 4, 8, 4, 2, 2, 3, 15, 3, 3, 6, 3, 7, 4, 10, 4, 8, 6, 4, 2, 2, 8, 4, 1, 35, 1, 4, 7, 4, 8, 6
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OFFSET
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1,1
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 8 = 24-16 where 16 is the first 4-almost prime and 24 is the second.
a(2) = 12 = 36-24.
a(3) = 4 = 40-36.
a(4) = 14 = 54-40.
a(5) = 2 = 56-54.
a(6) = 4 = 60-56.
a(7) = 21 = 81-60.
a(13) = 22 = 126-104.
a(21) = 28 = 184-156.
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MAPLE
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A114404 := proc(nmax) local a, i, a014613 ; a := [] ; i := 1 ; a014613 := -1 ; while nops(a) < nmax do if numtheory[bigomega](i) = 4 then if a014613 > 0 then a := [op(a), i-a014613] ; fi ; a014613 := i ; fi ; i := i+1 ; end: a ; end: A114404(200) ; # R. J. Mathar, May 10 2007
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MATHEMATICA
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Differences[Select[Range[800], Total[FactorInteger[#][[All, 2]]]==4&]] (* Harvey P. Dale, Feb 14 2017 *)
Select[Range[1000], PrimeOmega[#]==4&]//Differences (* Harvey P. Dale, May 12 2018 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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OFFSET
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1,1
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COMMENTS
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A109986 is primes ordered alphabetically by where they occur in A000052. A109987 is semiprimes ordered alphabetically by where they occur in A000052. A109988 is 3-almost primes ordered alphabetically by where they occur in A000052.
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LINKS
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EXAMPLE
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a(1) = 88 because eighty-eight is the first 4-almost prime in alphabetical order, there being no 1-digit 4-almost primes.
a(2) = 84 because eighty-four is the second 2-digit 4-almost prime in alphabetical order.
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CROSSREFS
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KEYWORD
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base,easy,nonn,word
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AUTHOR
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STATUS
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approved
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A146484
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Decimal expansion of Product_{q in A014613} (1-1/(q*(q-1))).
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+20
1
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9, 8, 9, 6, 2, 8, 8, 6, 7, 1, 6, 6, 4, 2, 7, 6, 6, 5, 5, 0, 4, 3, 2, 2, 8, 3, 7, 4, 5, 7, 9, 2, 4, 3, 0, 8, 0, 5, 7, 5, 5, 7, 5, 8, 9, 3, 5, 0, 2, 9, 6, 5, 3, 4, 8, 4, 4
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_4(s)/j at r=1, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
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EXAMPLE
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0.989628867166427665504.. = (1-1/240)*(1-1/552)*...
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KEYWORD
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AUTHOR
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STATUS
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approved
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A146488
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Decimal expansion of Product_{q in A014613} (1-1/(q^2*(q-1))).
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+20
1
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9, 9, 9, 5, 9, 5, 2, 7, 8, 5, 8, 6, 5, 3, 5, 5, 3, 5, 6, 3, 7, 4, 5, 2, 4, 9, 3, 2, 4, 8, 3, 3, 6, 4, 5, 3, 0, 8, 3, 6, 5, 0, 6, 3, 2, 4, 1, 2, 6, 7, 4, 0, 4, 9, 8, 8, 7
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_4(s)/j at r=2, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
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EXAMPLE
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0.99959527858653553563... = (1-1/3840)*(1-1/13248)*(1-1/45360)*(1-1/62400)*..
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KEYWORD
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AUTHOR
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STATUS
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approved
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89, 151, 197, 233, 307, 349, 461, 491, 569, 571, 739, 857, 859, 1013, 1061, 1097, 1277, 1291, 1303, 1483, 1667, 1747, 1831, 1913, 1973, 2003, 2131, 2357, 2503, 2531, 2621, 2683, 3011, 3067, 3163, 3209, 3229, 3259, 3271, 3581, 3797, 3929, 4013, 4027, 4073, 4219, 4327, 4597, 4793, 4877, 4903
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OFFSET
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1,1
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COMMENTS
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Primes p such that p - 1 and p + 1 are 4-almost primes.
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LINKS
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EXAMPLE
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89 - 1 = 2^3*11, 89 + 1 = 2*3^2*5.
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MATHEMATICA
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Select[Prime[Range[1000]], 4 == PrimeOmega[# - 1] == PrimeOmega[# + 1] &]
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PROG
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(PARI) forprime(p= 1, 5000, if(4==bigomega(p-1)&&4==bigomega(p+1), print1(p", ")))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A253919
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Indices of products of four distinct primes (A046386) in the sequence of products of 4 primes (A014613).
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+20
1
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27, 44, 56, 63, 71, 78, 83, 99, 103, 111, 115, 130, 133, 139, 140, 145, 166, 168, 171, 176, 185, 188, 190, 199, 201, 207, 208, 213, 217, 221, 229, 233, 239, 244, 248, 266, 271, 274, 276, 278, 285, 292, 299, 306, 313, 316, 317, 320, 322, 325, 331, 337, 341, 347, 351, 353, 357, 363, 375, 381, 387, 388, 389, 393, 394, 396, 402
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OFFSET
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1,1
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COMMENTS
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Or, positions of squarefree numbers in A014613.
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LINKS
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FORMULA
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EXAMPLE
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a(10000) = 25632 because A014613(25632) = A046386(100000) = 135555 = 2*3*7*1291 10000st squarefree number in A014613.
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MATHEMATICA
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c = 0; s = {}; Do[If[4 == PrimeOmega[k], c++; If[{1, 1, 1, 1} == (#[[2]] & /@ FactorInteger[k]) , AppendTo[s, c]]], {k, 16, 3000}]; s
(* or *) c = 0; s2 = {}; Do[If[4 == PrimeOmega[k], c++; If[SquareFreeQ[k] , AppendTo[s2, c]]], {k, 16, 3000}]; s2
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PROG
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(PARI) {c = 0; for (k = 16, 3000, if (4 == bigomega (k), c++; if (issquarefree (k), print1 (c ", "))))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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