A195382 Numbers such that the difference between the sum of the even divisors and the sum of the odd divisors is prime.

4, 8, 16, 18, 32, 50, 256, 512, 578, 1458, 2048, 3362, 4802, 6962, 8192, 10082, 15842, 20402, 31250, 34322, 55778, 57122, 59858, 167042, 171698, 293378, 524288, 559682, 916658, 982802, 1062882, 1104098, 1158242, 1195058, 1367858, 1407842, 1414562

Offset: 1

Author

Michel Lagneau, Sep 17 2011

Keywords

• nonn

Comments

Note that these are all even numbers. The odd numbers, producing the negative of a prime, are all squares whose square roots are in A193070. - T. D. Noe, Sep 19 2011

Examples

			The divisors of 18 are  { 1, 2, 3, 6, 9, 18}, and  (2 + 6 + 18) - (1 + 3 + 9) = 13  is prime. Hence 18 is in the sequence.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A088827.

Cf. A002129, A113184.

Programs

Maple

with(numtheory):for n from 2 by 2 to 200 do:x:=divisors(n):n1:=nops(x):s1:=0:s2:=0:for m from 1 to n1 do:if irem(x[m],2)=1 then s1:=s1+x[m]:else s2:=s2+x[m]:fi:od: if type(s2-s1,prime)=true then printf(`%d, `,n): else fi:od:

Mathematica

f[n_] := Module[{d = Divisors[n], p}, p = Plus @@ Select[d, OddQ] - Plus @@ Select[d, EvenQ]; PrimeQ[p]]; Select[Range[2,1000000,2], f] (* T. D. Noe, Sep 19 2011 *)

PARI

list(lim)=my(v=List(),t);forstep(n=3,sqrt(lim\2),2,if(isprime(s=sigma(n^2)),listput(v,2*n^2)));t=2;while((t*=2)<=lim,if(isprime(2*sigma(t/2)-1),listput(v,t)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 18 2011