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%I #16 Dec 06 2022 03:22:15
%S 0,3,4,5,6,14,8,7,7,18,12,22,14,22,20,9,18,23,20,28,24,30,24,30,11,34,
%T 10,34,30,52,32,11,32,42,28,36,38,46,36,38,42,60,44,46,32,54,48,38,15,
%U 31,44,52,54,32,36,46,48,66,60,80,62,70,38,13,40,76,68,64,56,68,72,49
%N a(n) = Sum_{p|n, p prime} Sum_{d|n} gcd(d,p).
%H Robert Israel, <a href="/A351711/b351711.txt">Table of n, a(n) for n = 1..10000</a>
%F a(p) = p + 1, p prime.
%F a(n) = tau(n)*omega(n) + Sum_{p|n, p prime} (p-1)*tau(n/p).
%e a(6) = 14; a(6) = Sum_{p|6, p prime} Sum_{d|6} gcd(d,p) = gcd(1,2) + gcd(2,2) + gcd(3,2) + gcd(6,2) + gcd(1,3) + gcd(2,3) + gcd(3,3) + gcd(6,3) = 1 + 2 + 1 + 2 + 1 + 1 + 3 + 3 = 14.
%p f:= proc(n) local p, P; uses numtheory;
%p P:= factorset(n);
%p tau(n)*nops(P)+add((p-1)*tau(n/p),p=P);
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Dec 05 2022
%o (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, (f[k,1]-1)*numdiv(n/f[k,1])) + omega(f)*numdiv(f); \\ _Michel Marcus_, Feb 18 2022
%Y Cf. A001221 (omega), A000005 (tau), A351758.
%K nonn,look
%O 1,2
%A _Wesley Ivan Hurt_, Feb 16 2022