cp's OEIS Frontend

A376886 The number of distinct factors of n of the form p^(k!), where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

%I A376886 #6 Oct 10 2024 15:34:34
%S A376886 0,1,1,1,1,2,1,2,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,3,1,2,2,2,1,3,1,2,2,2,
%T A376886 2,2,1,2,2,3,1,3,1,2,2,2,1,2,1,2,2,2,1,3,2,3,2,2,1,3,1,2,2,1,2,3,1,2,
%U A376886 2,3,1,3,1,2,2,2,2,3,1,2,1,2,1,3,2,2,2,3,1,3,2,2,2,2,2,3,1,2,2,2,1,3,1,3,3
%N A376886 The number of distinct factors of n of the form p^(k!), where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.
%C A376886 See A376885 for details about this factorization.
%C A376886 First differs from A371090 at n = 2^18 = 262144.
%C A376886 Differs from A064547 at n = 64, 128, 192, 256, 320, 384, 448, 512, ... .
%C A376886 Differs from A058061 at n = 128, 384, 512, 640, 896, ... .
%H A376886 Amiram Eldar, <a href="/A376886/b376886.txt">Table of n, a(n) for n = 1..10000</a>
%F A376886 Additive with a(p^e) = A060130(e).
%F A376886 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.12589120926760155013..., where f(x) = -x + (1-x) * Sum_{k>=1} A060130(k) * x^k.
%e A376886 For n = 8 = 2^3, the representation of 3 in factorial base is 11, i.e., 3 = 1! + 2!, so 8 = (2^(1!))^1 * (2^(2!))^1 and a(8) = 1 + 1 = 2.
%e A376886 For n = 16 = 2^4, the representation of 4 in factorial base is 20, i.e., 4 = 2 * 2!, so 16 = (2^(2!))^2 and a(16) = 1.
%t A376886 fdignum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, s++]; m++]; s]; f[p_, e_] := fdignum[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A376886 (PARI) fdignum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, if(r > 0, s ++); m++); s;}
%o A376886 a(n) = {my(e = factor(n)[, 2]); sum(i = 1, #e, fdignum(e[i]));}
%Y A376886 Cf. A007623, A058061, A060130, A077761, A371090.
%Y A376886 Similar sequences: A064547, A318464, A376885.
%K A376886 nonn,easy,base
%O A376886 1,6
%A A376886 _Amiram Eldar_, Oct 08 2024