A252040 Least number k such that the sum of the n-th powers of divisors of k is prime, or 0 if there is no such k.

2, 2, 4, 2, 3418801, 0, 64, 2, 4, 0, 21611482313284249, 0, 38580382095460899155325996786287338133521, 0, 0, 2, 2908327322588596409721563830760333292497745890881, 0, 366742604046618684582110328334808145525757868881, 0, 0, 0, 264638735999480827156185738343112840094453729311824181089

Offset: 1

Author

Derek Orr, Dec 12 2014

Keywords

• nonn

Comments

a(n) = 2 if and only if 2^n + 1 is in A019434.

From Jinyuan Wang, Jan 30 2021: (Start)

a(n) = 0 if n > 1 is not a prime power. Proof: note that sigma_n(k) = Product_{i=1..m} (1 + p_i^n + ... + p_i^(n*e_i)), where k = Product_{i=1..m} p_i^e_i. We only need to prove when n > 1 is not a prime power and e > 1, s = Sum_{i=0..e-1} p^(n*i) = (p^(n*e) - 1)/(p^n - 1) is composite. If e is prime, then s is divisible by (p^(e^(t+1)) - 1)/(p^(e^t) - 1), where t is the e-adic valuation of n. If e is composite, then s is divisible by (p^(n*q) - 1)/(p^n - 1), where q is a prime factor of e.

Corollary: k must be of the form p^(e - 1) when n = e^t, where p and e are primes. Therefore, a(2^t) = 0 if 2^2^t + 1 is composite. (End)

Examples

			2 has two divisors, 2 and 1. 2^3 + 1^3 = 9 is not prime.
3 has two divisors, 3 and 1. 3^3 + 1^3 = 28 is not prime.
4 has three divisors, 4, 2, and 1. 4^3 + 2^3 + 1^3 = 73 is prime. So, a(3) = 4.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..50

Crossrefs

Cf. A000203, A001157, A001158, A001159, A001160, A019434.

Cf. A023194 (sigma(n) is prime), A063783 (sigma_3(n) is prime).

Programs

Mathematica

a252040[n_Integer] := If[PrimePowerQ[n] && (p=First@ First@ FactorInteger[n])>2, q=2; While[!PrimeQ[DivisorSigma[n, q^(p-1)]], q=NextPrime[q]]; q^(p-1), 2*Boole[PrimeQ[2^n+1]]]; a252040 /@ Range[10] (* Michael De Vlieger, Dec 13 2014 *) (* modified by Jinyuan Wang, Jan 30 2021 *)

PARI

a(n) = if(isprimepower(n, &p) && p>2, my(q=2); while(!ispseudoprime(sigma(q^(p-1), n)), q=nextprime(q+1)); q^(p-1), 2*isprime(2^n+1)); \\ Modified by Jinyuan Wang, Jan 25 2021

Extensions

Name edited by and more terms from Jinyuan Wang, Jan 30 2021