A278911 Odd numbers with prime sum of divisors.

9, 25, 289, 729, 1681, 2401, 3481, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 83521, 85849, 146689, 279841, 458329, 491401, 531441, 552049, 579121, 597529, 683929, 703921, 707281, 734449, 829921, 1190281, 1203409, 1352569, 1394761, 1423249, 1481089

Offset: 1

Author

Jaroslav Krizek, Nov 30 2016

Keywords

• nonn

Comments

Also odd numbers with prime number and sum of divisors; if the sum of divisors is prime, then the number of divisors is prime.

Values of prime sums are sorted in A247837.

Subsequence of A050150 (odd numbers with prime number of divisors).

Odd terms of A023194.

All terms are squares of the form p^e such that p is odd prime and e+1 is a prime.

Examples

			sigma(9) = 13 (prime).
		

Links

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

Crossrefs

Cf. A000203, A005408, A023194, A050150, A193070, A247837, A278914.

Programs

Magma

[n: n in[2..10^7] | IsOdd(n) and IsPrime(SumOfDivisors(n)) and IsPrime(NumberOfDivisors(n))];

Maple

N:= 10^7: # to get all terms <= N
Ps:= select(isprime, [seq(i,i=3..floor(N^(1/2)),2)]):
es:= map(`-`,select(isprime, [seq(i,i=3..floor(log[3](N))+1,2)]),1):
Pes:= [seq(seq([p,e],p=Ps),e=es)]:
filter:= proc(pe) local v; v:= (pe[1]^(pe[2]+1)-1)/(pe[1]-1); pe[1]^pe[2] <= N and isprime(v) end proc:
sort(map(pe -> pe[1]^pe[2], select(filter, Pes))); # Robert Israel, Jan 22 2019

Mathematica

Select[Range[1, 2*10^6, 2], PrimeQ@DivisorSigma[1, #] &] (* Michael De Vlieger, Dec 01 2016 *)

PARI

isok(n) = (n % 2) && isprime(sigma(n)); \\ Michel Marcus, Dec 01 2016

Formula

a(n) = A193070(n)^2. - Michel Marcus, Dec 01 2016