A275598 Primes p such that the number of odd divisors of p-1 is a prime q which is equal to the number of odd divisors of p+1.

11, 13, 23, 47, 193, 383, 577

Offset

1,1

Comments

Conjecture: this sequence is finite.

Any further terms are greater than 10^10. - Charles R Greathouse IV, Aug 22 2016

Any further terms are greater than 2 * 10^12. - Dana Jacobsen, Aug 30 2016

Links

Table of n, a(n) for n=1..7.

Example

11 is in this sequence because there are 2 odd divisors 1 and 5 of 10 and there are 2 odd divisors 1 and 3 of 12, and 2 is a prime.

Maple

filter:= proc(p) local r, q;
   r:= numtheory:-tau((p-1)/2^padic:-ordp(p-1, 2));
   if not isprime(r) then return false fi;
   r = numtheory:-tau((p+1)/2^padic:-ordp(p+1, 2))
end proc:
res:= NULL: p:= 0:
while p < 1000 do
  p:= nextprime(p);
  if filter(p) then
    res:= res, p;
  fi;
od:
res; # Robert Israel, Aug 24 2016

Mathematica

okQ[p_?PrimeQ] := Module[{r}, r = DivisorSigma[0, (p-1)/2^IntegerExponent[p-1, 2]]; If[!PrimeQ[r], Return[False]]; r == DivisorSigma[0, (p+1)/2^IntegerExponent[p+1, 2]]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Feb 09 2023, after Robert Israel *)

Prog

(Perl) use ntheory ":all"; forprimes { $n1 = scalar(grep { $_&1 } divisors($_-1)); say if is_prime($n1) && $n1 == scalar(grep { $_&1 } divisors($_+1)); } 1e7; # Dana Jacobsen, Aug 24 2016
(PARI) f(n)=numdiv(n>>valuation(n, 2))
is(n)=if(!isprime(n), return(0)); my(q=f(n-1)); isprime(q) && f(n+1)==q \\ Charles R Greathouse IV, Aug 24 2016

Crossrefs

Cf. A001227, A275418.

Sequence in context: A127043 A084952 A277048 * A090433 A022325 A295340

Adjacent sequences: A275595 A275596 A275597 * A275599 A275600 A275601

Keyword

nonn

Author

Juri-Stepan Gerasimov, Aug 23 2016

Status

approved