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A336562 Number of pairs sigma(p^x), sigma(q^y) that are not coprime, where p^x and q^y are any two maximal prime power divisors of n, with p < q. a(1) = 0.

%I A336562 #18 Jul 28 2020 02:06:40
%S A336562 0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,2,0,0,1,1,
%T A336562 1,0,0,0,1,1,0,1,0,0,0,1,0,0,0,0,1,1,0,0,1,0,1,1,0,1,0,0,0,0,1,2,0,0,
%U A336562 1,2,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,2,0,0,3
%N A336562 Number of pairs sigma(p^x), sigma(q^y) that are not coprime, where p^x and q^y are any two maximal prime power divisors of n, with p < q. a(1) = 0.
%H A336562 Antti Karttunen, <a href="/A336562/b336562.txt">Table of n, a(n) for n = 1..65537</a>
%H A336562 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A336562 For n = 40 = 2^3 * 5^1, sigma(2^3) = 15 and sigma(5) = 6, so we have one such prime power pair that the gcd of their sum of divisors is larger than one (in this case gcd(15,6) = 3), thus a(40) = 1.
%e A336562 For n = 120 = 2^3 * 3^1 * 5^1, possible pairs are [sigma(8), sigma(3)], [sigma(8), sigma(5)] and [sigma(3), sigma(5)], with gcd(15,4) = 1, gcd(15,6) = 3 and gcd(4,6) = 2, thus there are two pairs that are not coprime, and a(120) = 2.
%o A336562 (PARI) A336562(n) = if(1==n,0,my(f=factor(n),s=0); for(i=1,#f~,for(j=1+i,#f~,if(1!=gcd(sigma(f[i,1]^f[i,2]),sigma(f[j,1]^f[j,2])), s++))); (s));
%Y A336562 Cf. A000203, A051027, A324892, A336355, A336356, A336546, A336547 (positions of zeros), A336548 (of nonzero terms).
%K A336562 nonn
%O A336562 1,30
%A A336562 _Antti Karttunen_, Jul 27 2020