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A384423 The number of prime powers (not including 1) p^e that divide n such that e is unitarily coprime to the p-adic valuation of n.

%I A384423 #8 May 28 2025 10:51:51
%S A384423 0,1,1,1,1,2,1,2,1,2,1,2,1,2,2,3,1,2,1,2,2,2,1,3,1,2,2,2,1,3,1,4,2,2,
%T A384423 2,2,1,2,2,3,1,3,1,2,2,2,1,4,1,2,2,2,1,3,2,3,2,2,1,3,1,2,2,2,2,3,1,2,
%U A384423 2,3,1,3,1,2,2,2,2,3,1,4,3,2,1,3,2,2,2
%N A384423 The number of prime powers (not including 1) p^e that divide n such that e is unitarily coprime to the p-adic valuation of n.
%C A384423 A number k is unitarily coprime to m if the largest divisor of k that is a unitary divisor of m is 1.
%H A384423 Amiram Eldar, <a href="/A384423/b384423.txt">Table of n, a(n) for n = 1..10000</a>
%F A384423 Additive with a(p^e) = uphi(e), where uphi is the unitary totient function (A047994).
%F A384423 Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A047994(e)/p^e = 0.74335242036929441969... .
%t A384423 f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n];
%t A384423 ff[p_, e_] := uphi[e]; a[1] = 0; a[n_] := Plus @@ ff @@@ FactorInteger[n]; Array[a, 100]
%o A384423 (PARI) uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1);}
%o A384423 a(n) = vecsum(apply(uphi, factor(n)[, 2]));
%Y A384423 Cf. A047994, A077761, A085548, A321167.
%K A384423 nonn,easy
%O A384423 1,6
%A A384423 _Amiram Eldar_, May 28 2025