A280590 Numbers k such that for any positive integers (a, b), if a * b = k then sigma(a) + sigma(b) is a prime.

1, 3, 5, 6, 11, 17, 24, 26, 27, 29, 38, 41, 59, 71, 101, 107, 125, 137, 149, 158, 179, 191, 197, 206, 218, 227, 239, 269, 281, 311, 344, 347, 419, 431, 446, 458, 461, 521, 536, 569, 599, 617, 641, 659, 698, 809, 821, 827, 857, 878, 881, 1019, 1031, 1049, 1061

Offset: 1

Author

Michel Lagneau, Jan 06 2017

Keywords

• nonn

Comments

The subsequence of primes {3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, ... is exactly A001359 (lesser of twin primes).

Examples

			1 is in the sequence because 1 = 1*1 and sigma(1) + sigma(1) = 1 + 1 = 2 is prime.
24 is in the sequence because A038548(24) = 4 => four decompositions of 24 = 1*24 = 2*12 = 3*8 = 4*6 and
sigma(1) + sigma(24) =  1 + 60 = 61 is prime;
sigma(2) + sigma(12) =  3 + 28 = 31 is prime;
sigma(3) + sigma(8)  =  4 + 15 = 19 is prime;
sigma(4) + sigma(6)  =  7 + 12 = 19 is prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A001359, A038548, A080715.

Programs

Mathematica

t={};Do[ds=Divisors[n];If[EvenQ[Length[ds]],ok=True;k=1;While[k<=Length[ds]/2&&(ok=PrimeQ[DivisorSigma[1,ds[[k]]]+DivisorSigma[1,ds[[-k]]]]),k++];If[ok,AppendTo[t,n]]],{n,2,4000}];t

PARI

isok(n) = {fordiv(n, d, if (d^2 <= n, if (! isprime(sigma(d) + sigma(n/d)), return (0)););); return(1);} \\ Michel Marcus, Jan 06 2017