cp's OEIS Frontend

A376887 The number of divisors of n that are products of factors of the form p^(k!) with multiplicities not larger than their multiplicity in n, where p is a prime and k >= 1, when the factorization of n is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

%I A376887 #8 Oct 10 2024 15:34:46
%S A376887 1,2,2,2,2,4,2,4,2,4,2,4,2,4,4,3,2,4,2,4,4,4,2,8,2,4,4,4,2,8,2,6,4,4,
%T A376887 4,4,2,4,4,8,2,8,2,4,4,4,2,6,2,4,4,4,2,8,4,8,4,4,2,8,2,4,4,2,4,8,2,4,
%U A376887 4,8,2,8,2,4,4,4,4,8,2,6,3,4,2,8,4,4,4
%N A376887 The number of divisors of n that are products of factors of the form p^(k!) with multiplicities not larger than their multiplicity in n, where p is a prime and k >= 1, when the factorization of n is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.
%C A376887 See A376885 for details about this factorization.
%C A376887 If n = Product p_i^e_i is the canonical prime factorization of n, then the divisors that are counted by this function are d = Product p_i^s_i, where all the digits of s_i in factorial base are not larger than the corresponding digits of e_i.
%C A376887 The sum of these divisors is given by A376888(n).
%H A376887 Amiram Eldar, <a href="/A376887/b376887.txt">Table of n, a(n) for n = 1..10000</a>
%F A376887 Multiplicative with a(p^e) = A227154(e).
%e A376887 For n = 12 = 2^2 * 3^1, the representation of 2 in factorial base is 10, i.e., 2 = 2!, so 12 = (2^(2!))^1 * (3^(1!))^1 and a(12) = (1+1) * (1+1) = 4, corresponding to the 4 divisors 1, 3, 4 and 12.
%t A376887 fdigprod[n_] := Module[{k = n, m = 2, r, s = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s *= (r+1); m++]; s]; f[p_, e_] := fdigprod[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A376887 (PARI) fdigprod(n) = {my(k = n, m = 2, r, s = 1); while([k, r] = divrem(k, m); k != 0 || r != 0, s *= (r+1); m++); s;}
%o A376887 a(n) = {my(e = factor(n)[, 2]); prod(i = 1, #e, fdigprod(e[i]));}
%Y A376887 Cf. A007623, A034968, A376885, A376886, A376888.
%Y A376887 Similar sequences: A037445, A038148, A074848, A318465, A331109.
%K A376887 nonn,easy,mult,base
%O A376887 1,2
%A A376887 _Amiram Eldar_, Oct 08 2024