A285015 -id:A285015 - cp's OEIS frontend

A175768 Primes of the form k * b^b + 1, with b > 1.

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 163, 173, 181, 193, 197, 229, 233, 241, 257, 269, 271, 277, 281, 293, 313, 317, 337, 349, 353, 373, 379, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487, 509, 521, 541, 557, 569, 577, 593, 601, 613

Offset: 1

Kevin Batista (kevin762401(AT)yahoo.com), Sep 01 2010

Without the restriction on b, the sequence would be identical to A000040.

			For a(3), 4 * 2^2 + 1 = 17, which is prime.
From _Seiichi Manyama_, Mar 27 2018: (Start)
   n | a(n)
  ---+----------------------------------
   1 |   5 =  1 * 2^2 + 1.
   2 |  13 =  3 * 2^2 + 1.
   3 |  17 =  4 * 2^2 + 1.
   4 |  29 =  7 * 2^2 + 1.
   5 |  37 =  9 * 2^2 + 1.
   6 |  41 = 10 * 2^2 + 1.
   7 |  53 = 13 * 2^2 + 1.
   8 |  61 = 15 * 2^2 + 1.
   9 |  73 = 18 * 2^2 + 1.
  10 |  89 = 22 * 2^2 + 1.
  11 |  97 = 24 * 2^2 + 1.
  12 | 101 = 25 * 2^2 + 1.
  13 | 109 = 27 * 2^2 + 1 = 4 * 3^3 + 1. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000040, A002144, A180362, A285015.

Take[ Select[ Union@ Flatten@ Table[ k*b^b + 1, {b, 2, 20}, {k, 148}], PrimeQ], 55] (* Robert G. Wilson v, Sep 01 2010 *)

isA175768(n)=if(!isprime(n),return(0)); if(n%4==1||n%27==1,return(1)); forprime(b=5,log(n)/log(7),if(n%(b^b)==1,return(1)));0 \\ Charles R Greathouse IV, Sep 02 2010

Corrected and edited by Charles R Greathouse IV, Sep 02 2010

A285016 Primes of the form p*b^b - 1, where p is a prime and b>1.

Original entry on oeis.org

7, 11, 19, 43, 53, 67, 163, 211, 283, 331, 523, 547, 691, 787, 907, 1051, 1123, 1171, 1279, 1531, 1723, 1867, 2011, 2083, 2251, 2347, 2371, 2467, 2707, 2731, 2803, 2971, 3187, 3307, 3547, 3643, 3907, 3931, 4051, 4243, 4363, 4603, 4651, 4723, 5107, 5227

Offset: 1

Vincenzo Librandi, May 12 2017

			a(1) = 2*(2^2)-1 = 7.
a(2) = 3*(2^2)-1 = 11.
a(3) = 5*(2^2)-1 = 19.
a(4) = 11*(2^2)-1 = 43.

Vincenzo Librandi, Table of n, a(n) for n = 1..2300

Cf. A000312, A090866, A285015.

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&]

is(n)=for(b=2,oo, my(B=b^b); if((n+1)%B==0 && isprime((n+1)/B), return(isprime(n))); if(2*B+1>n, return(0))) \\ Charles R Greathouse IV, Jun 16 2022

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

A286658 Primes of the form p*b^b + 1, where p is a prime and b>1.

Original entry on oeis.org

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

Offset: 1

Vincenzo Librandi, May 12 2017

			a(1) = 3*(2^2)+1 = 13.
a(2) = 7*(2^2)+1 = 29.
a(3) = 13*(2^2)+1 = 53.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A175768, A285015.

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:
sort(convert({Res},list)); # Robert Israel, Nov 12 2019

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&]

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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A175768 Primes of the form k * b^b + 1, with b > 1.

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A285016 Primes of the form p*b^b - 1, where p is a prime and b>1.

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A286658 Primes of the form p*b^b + 1, where p is a prime and b>1.

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