A333926 The sum of recursive divisors of n.

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144

Offset

1,2

Comments

The definition of recursive divisors and the number of recursive divisors of n are in A282446.

First differs from A051378 at n = 256.

Links

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

Formula

Multiplicative with a(p^k) = 1 + Sum_{d recursive divisor of k} p^d.

a(n) <= A051378(n) <= A000203(n).

Example

The recursive divisors of 8 are 1, 2 and 8, therefore a(8) = 1 + 2 + 8 = 11.

Mathematica

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100]

Crossrefs

Cf. A000203, A051378, A282446, A287957, A287958.

Sequence in context: A353900 A366903 A049418 * A051378 A366440 A344575

Adjacent sequences: A333923 A333924 A333925 * A333927 A333928 A333929

Keyword

nonn,mult

Author

Amiram Eldar, Apr 10 2020

Status

approved