A376888 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.

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, 42, 40, 40, 30, 72, 32, 63, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 84, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144

Offset

1,2

Comments

See A376885 for details about this factorization.

First differs from A188999 at n = 16.

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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).

Example

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.

Mathematica

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]

Prog

(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; }
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))); }

Crossrefs

Cf. A188999, A376885, A376886, A376887.

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

Sequence in context: A327668 A324706 A049417 * A331110 A188999 A186644

Adjacent sequences: A376885 A376886 A376887 * A376889 A376890 A376891

Keyword

nonn,easy,mult,base

Author

Amiram Eldar, Oct 08 2024

Status

approved