login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Search: a014613 -id:a014613
Displaying 1-10 of 143 results found. page 1 2 3 4 5 6 7 8 9 10 ... 15
     Sort: relevance | references | number | modified | created      Format: long | short | data
A101638 Number of distinct 4-almost primes A014613 which are factors of n. +20
15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,48
COMMENTS
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).
REFERENCES
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.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. A. Bender and J. R. Goldman, On the Applications of Moebius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, 789-803, 1975.
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 arXiv version]
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]
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Moebius Transform..
EXAMPLE
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.
PROG
(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
CROSSREFS
Cf. A101638, A014613, A000332, A086971, A100605, A000040, A001358, A014612, A014614.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Dec 10 2004
STATUS
approved
A109024 Numbers that have exactly four prime factors counted with multiplicity (A014613) whose digit reversal is different and also has 4 prime factors (with multiplicity). +20
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This sequence is the k = 4 instance of the series which begins with k = 1, k = 2, k = 3 (A109023).
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.
EXAMPLE
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).
MATHEMATICA
Select[Range[2226], PrimeOmega[#]==4 && PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==4 &&!PalindromeQ[#]&] (* James C. McMahon, Mar 07 2024 *)
PROG
(PARI) is(n) = {
my(r = fromdigits(Vecrev(digits(n))));
n!=r && bigomega(n) == 4 && bigomega(r) == 4
} \\ David A. Corneth, Mar 07 2024
CROSSREFS
Cf. A006567, A097393, A109018, A109023, A109025, A109026, A109027, A109028, A109029, A109030, A109031.
KEYWORD
nonn,base
AUTHOR
Jonathan Vos Post, Jun 16 2005
STATUS
approved
A124284 Prime(4almostprime(n))-4almostprime(prime(n)). Commutator [A000040,A014613] at n. +20
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Robert G. Wilson v, Table of n, a(n) for n=1..1000
FORMULA
a(n) = prime(4almostprime(n)) - 4almostprime(prime(n)) = A000040(A014613(n)) -A014613(A000040(n)).
EXAMPLE
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.
MATHEMATICA
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}]
PROG
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot, prime
def A124284(n):
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
return prime(m)-r # Chai Wah Wu, Aug 17 2024
CROSSREFS
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)).
Cf. A000040, A014613, A114414, A122824, A124269, A124270, A124282, A124283.
KEYWORD
easy,nonn,changed
AUTHOR
Jonathan Vos Post, Oct 24 2006
EXTENSIONS
More terms from Robert G. Wilson v, Aug 31 2007
STATUS
approved
A114426 Product of the first n 4-almost primes (A014613). +20
4
16, 384, 13824, 552960, 29859840, 1672151040, 100329062400, 8126654054400, 682638940569600, 60072226770124800, 5406500409311232000, 540650040931123200000, 56227604256836812800000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..13.
FORMULA
a(n) = Prod[from i = 1 to n] A014613(i).
EXAMPLE
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).
MATHEMATICA
FoldList[Times, Select[Range[200], PrimeOmega[#]==4&]] (* Harvey P. Dale, Dec 02 2018 *)
CROSSREFS
Cf. A001222, A002110, A008585, A014613, A008586, A086046, A112141.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 13 2006
STATUS
approved
A114404 4-almost prime gaps. First differences of A014613. +20
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n=1..10000
FORMULA
a(n) = A014613(n+1) - A014613(n).
EXAMPLE
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.
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Cf. A014613, A065516, A111870, A111871, A114403-A114411, A114412-A114422.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Nov 25 2005
EXTENSIONS
Corrected and extended by R. J. Mathar, May 10 2007
STATUS
approved
A109989 4-almost primes (A014613) ordered alphabetically by where they occur in A000052. +20
1
88, 84, 81, 54, 56, 40, 90 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..7.
EXAMPLE
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.
CROSSREFS
Cf. A000052, A014613, A109986, A109987, A109988.
KEYWORD
base,easy,nonn,word
AUTHOR
Jonathan Vos Post, Jul 07 2005
STATUS
approved
A146484 Decimal expansion of Product_{q in A014613} (1-1/(q*(q-1))). +20
1
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
4-almost prime analog of A005596.
LINKS
Table of n, a(n) for n=0..56.
FORMULA
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.
EXAMPLE
0.989628867166427665504.. = (1-1/240)*(1-1/552)*...
KEYWORD
nonn,cons,less
AUTHOR
R. J. Mathar, Feb 13 2009
STATUS
approved
A146488 Decimal expansion of Product_{q in A014613} (1-1/(q^2*(q-1))). +20
1
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
4-almost prime analog of A065414.
LINKS
Table of n, a(n) for n=0..56.
FORMULA
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.
EXAMPLE
0.99959527858653553563... = (1-1/3840)*(1-1/13248)*(1-1/45360)*(1-1/62400)*..
KEYWORD
nonn,cons,less
AUTHOR
R. J. Mathar, Feb 13 2009
STATUS
approved
A247114 Primes sandwiched between 4-almost primes (A014613). +20
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Primes p such that p - 1 and p + 1 are 4-almost primes.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
89 - 1 = 2^3*11, 89 + 1 = 2*3^2*5.
MATHEMATICA
Select[Prime[Range[1000]], 4 == PrimeOmega[# - 1] == PrimeOmega[# + 1] &]
PROG
(PARI) forprime(p= 1, 5000, if(4==bigomega(p-1)&&4==bigomega(p+1), print1(p", ")))
(PARI) is(n)=bigomega(n-1)==4 && bigomega(n+1)==4 && isprime(n) \\ Charles R Greathouse IV, Apr 27 2015
CROSSREFS
Cf. A014612, A014613, A046314, A154598, A063644, A247088.
KEYWORD
nonn
AUTHOR
Zak Seidov, Jan 10 2015
STATUS
approved
A253919 Indices of products of four distinct primes (A046386) in the sequence of products of 4 primes (A014613). +20
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Or, positions of squarefree numbers in A014613.
LINKS
Zak Seidov, Table of n, a(n) for n = 1..10000
FORMULA
A014613(a(n))=A046386(n).
EXAMPLE
a(1) = 27 because A014613(27) = A046386(1) = 210 = 2*3*5*7 the first squarefree number in A014613.
a(10000) = 25632 because A014613(25632) = A046386(100000) = 135555 = 2*3*7*1291 10000st squarefree number in A014613.
MATHEMATICA
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
PROG
(PARI) {c = 0; for (k = 16, 3000, if (4 == bigomega (k), c++; if (issquarefree (k), print1 (c ", "))))}
CROSSREFS
Cf. A014613, A046386.
KEYWORD
nonn
AUTHOR
Zak Seidov, Jan 19 2015
STATUS
approved
page 1 2 3 4 5 6 7 8 9 10 ... 15

Search completed in 0.060 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 25 14:15 EDT 2024. Contains 375439 sequences. (Running on oeis4.)