A165320 -id:A165320 - cp's OEIS frontend

A165318 Primes p where the number of divisors of p-1 is a power of 2.

2, 3, 7, 11, 23, 31, 41, 43, 47, 59, 67, 71, 79, 83, 89, 103, 107, 131, 137, 139, 167, 179, 191, 211, 223, 227, 233, 239, 251, 263, 271, 281, 283, 311, 313, 331, 347, 359, 367, 379, 383, 409, 419, 431, 439, 443, 457, 463, 467, 479, 499, 503, 521, 547, 563, 569

Offset: 1

Leroy Quet, Sep 14 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A008328, A036537, A165319, A165320.

isA000079 := proc(n) RETURN( n=1 or numtheory[factorset](n) = {2}) ; end: A165318 := proc(n) if n = 1 then 2; else p := nextprime(procname(n-1)) ; while not isA000079(numtheory[tau](p-1)) do p := nextprime(p) ; od; p ; fi; end: seq(A165318(n),n=1..90) ; # R. J. Mathar, Sep 18 2009

Select[Prime[Range[200]],IntegerQ[Log[2,DivisorSigma[0,#-1]]]&] (* Harvey P. Dale, Oct 14 2018 *)

isok(p) = isprime(p) && apply(x -> x >> valuation(x, 2), numdiv(p-1)) == 1; \\ Amiram Eldar, Jun 26 2025

More terms from R. J. Mathar, Sep 18 2009

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

Original entry on oeis.org

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

Leroy Quet, Sep 14 2009

nonn

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

Cf. A165318, A165320.

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

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

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 *)

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

Extended by R. J. Mathar, Sep 20 2009

cp's OEIS Frontend

A165318 Primes p where the number of divisors of p-1 is a power of 2.

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