A165319 Primes p where the number of divisors of p+1 is a power of 2.

2, 5, 7, 13, 23, 29, 37, 41, 53, 61, 73, 101, 103, 109, 113, 127, 137, 151, 157, 167, 173, 181, 193, 229, 257, 263, 269, 277, 281, 311, 313, 317, 353, 373, 383, 389, 397, 401, 409, 421, 433, 439, 457, 461, 487, 509, 541, 569, 593, 601, 613, 617, 631, 641, 653

Offset: 1

Author

Leroy Quet, Sep 14 2009

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A165318, A165320.

Cf. A008329. [R. J. Mathar, Sep 20 2009]

Programs

Maple

b:= proc(n) option remember; is(n=2^ilog2(n)) end:
a:= proc(n) option remember; local p; p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if andmap(b, map(i-> i[2]+1, ifactors(p+1)[2])) then break fi
      od; p
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, Sep 05 2019

Mathematica

okQ[p_] := PrimeQ[p] && IntegerQ[Log[2, DivisorSigma[0, p+1]]];
Select[Prime[Range[200]], okQ] (* Jean-François Alcover, May 20 2020 *)
Select[Prime[Range[150]],IntegerQ[Log2[DivisorSigma[0,#+1]]]&] (* Harvey P. Dale, Jul 30 2025 *)

PARI

isok(p) = isprime(p) && (nd = numdiv(p+1)) && (nd == 2^valuation(nd, 2)); \\ Michel Marcus, Sep 05 2019

Extensions

Extended by R. J. Mathar, Sep 20 2009