Search: a175768 -id:a175768
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A285015
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Primes of the form k * b^b - 1, with b > 1.
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+10
4
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3, 7, 11, 19, 23, 31, 43, 47, 53, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 269, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547
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internal format)
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OFFSET
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1,1
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COMMENTS
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This sequence has relative density 1 - Prod_p 1-1/(p^p-p) = 0.52098749404893... in the primes, hence a(n) ~ kn log n where k = 1.91943... is the reciprocal of this quantity. - Charles R Greathouse IV, May 12 2017
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LINKS
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EXAMPLE
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a(1) = 1*(2^2)-1 = 3.
a(2) = 2*(2^2)-1 = 7.
a(9) = 2*(3^3)-1 = 53.
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MAPLE
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N:= 1000: # to get all terms <= N
bmax:= floor(ln(N+1)/LambertW(ln(N+1))):
sort(convert(select(isprime, {seq(seq(k*b^b-1, k=1..(N+1)/b^b), b=2..bmax)}), list)); # Robert Israel, May 11 2017
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MATHEMATICA
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Take[Select[Union@Flatten@Table[k b^b - 1, {b, 2, 20}, {k, 148}], PrimeQ], 55]
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PROG
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(PARI) upto(n)=my(l=List([3]), b=2, s=1); n++; while(b^b < n, c = b^b; forstep(i=2, n\c, s, if(isprime(i*c-1), listput(l, i*c-1))); s=3-s; b++); listsort(l, 1); l \\ David A. Corneth, May 11 2017
(PARI) is(n)=if(!isprime(n), return(0)); my(t); forprime(p=2, , t=p^p; if((n+1)%t==0, return(1)); if(t>=n, return(0))) \\ Charles R Greathouse IV, May 11 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A235467
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Primes whose base-4 representation also is the base-3 representation of a prime.
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+10
2
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2, 89, 137, 149, 281, 293, 353, 389, 409, 421, 593, 613, 661, 1097, 1109, 1289, 1301, 1321, 1381, 1409, 1601, 1609, 1669, 2069, 2129, 2309, 2377, 2389, 2729, 4133, 4229, 4373, 4441, 4513, 4673, 5153
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OFFSET
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1,1
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COMMENTS
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This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
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LINKS
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EXAMPLE
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E.g., 89 = 1121_4 and 1121_3 = 43 both are prime.
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MATHEMATICA
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b4b3Q[n_]:=Module[{b4=IntegerDigits[n, 4]}, Max[b4]<3&&PrimeQ[ FromDigits[ b4, 3]]]; Select[Prime[Range[700]], b4b3Q] (* Harvey P. Dale, Dec 14 2021 *)
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PROG
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(PARI) is(p, b=3, c=4)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)
(PARI) forprime(p=1, 1e3, is(p, 4, 3)&&print1(vector(#d=digits(p, 3), i, 4^(#d-i))*d~, ", ")) \\ To produce the terms, this is more efficient than to select them using straightforwardly is(.)=is(., 3, 4)
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CROSSREFS
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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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A286658
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Primes of the form p*b^b + 1, where p is a prime and b>1.
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+10
1
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13, 29, 53, 149, 173, 269, 293, 317, 389, 509, 557, 653, 769, 773, 797, 1109, 1229, 1493, 1637, 1733, 1949, 1997, 2309, 2477, 2693, 2837, 2909, 2957, 3329, 3413, 3533, 3677, 3989, 4133, 4157, 4253, 4349, 4373, 4493, 4517, 5189, 5309, 5693, 5717, 5813, 6173
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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LINKS
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EXAMPLE
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a(1) = 3*(2^2)+1 = 13.
a(2) = 7*(2^2)+1 = 29.
a(3) = 13*(2^2)+1 = 53.
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MAPLE
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N:= 10000: # for all terms <= N
Res:= NULL:
P:= select(isprime, [2, seq(i, i=3..N/4, 2)]):
for b from 2 do
q:= b^b; if q > N/2 then break fi;
for i from 1 to nops(P) do
x:= P[i]*q+1;
if x > N then break fi;
if isprime(x) then Res:= Res, x fi;
od od:
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MATHEMATICA
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nmax=10^4; pimax=PrimePi[nmax]; bmax=1; While[(bmax+1)^(bmax+1)<=nmax, bmax++]; Select[Union@Flatten@Table[Prime[pi] b^b+1, {b, 2, bmax}, {pi, pimax}], PrimeQ[#]&&#<=nmax&]
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PROG
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(PARI) list(lim)=my(v=List()); lim\=1; for(b=2, oo, my(p=2*b^b+1); if(p>lim, break); if(isprime(p), listput(v, p))); forstep(b=2, oo, 2, my(B=b^b); if(3*B+1>lim, break); forprime(q=3, (lim-1)\B, my(p=q*B+1); if(isprime(p), listput(v, p)))); Set(v) \\ Charles R Greathouse IV, Jun 16 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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