A376888 - OEIS

%I #7 Oct 10 2024 15:34:55

%S 1,3,4,5,6,12,8,15,10,18,12,20,14,24,24,21,18,30,20,30,32,36,24,60,26,

%T 42,40,40,30,72,32,63,48,54,48,50,38,60,56,90,42,96,44,60,60,72,48,84,

%U 50,78,72,70,54,120,72,120,80,90,60,120,62,96,80,65,84,144

%N The sum 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 See A376885 for details about this factorization.

%C First differs from A188999 at n = 16.

%C The number of these divisors is given by A376887(n).

%H Amiram Eldar, <a href="/A376888/b376888.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative: if e = Sum_{k>=1} d_k * k! (factorial base representation), then a(p^e) = Product_{k>=1} (p^(k!*{d_k+1}) - 1)/(p^(k!) - 1).

%e 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) is the sum of the 4 divisors 1 + 3 + 4 + 12 = 20.

%t ff[q_, s_] := (q^(s + 1) - 1)/(q - 1); f[p_, e_] := Module[{k = e, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, AppendTo[s, {p^(m - 1)!, r}];]; m++]; Times @@ ff @@@ s]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) fdigits(n) = {my(k = n, m = 2, r, s = []); while([k, r] = divrem(k, m); k != 0 || r != 0, s = concat(s, r); m++); s;}

%o a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2], d); prod(i = 1, #p, prod(j = 1, #d=fdigits(e[i]), (p[i]^(j!*(d[j]+1)) - 1)/(p[i]^j! - 1)));}

%Y Cf. A188999, A376885, A376886, A376887.

%Y Similar sequences: A049417, A049418, A074847, A097863, A331107, A331110.

%K nonn,easy,mult,base

%O 1,2

%A _Amiram Eldar_, Oct 08 2024