A336465 Recursive variant of A057723: If n is cubefree, then a(n) = A057723(n), otherwise a(n) = A007947(n) * a(n/A007947(n)).

1, 2, 3, 6, 5, 6, 7, 12, 12, 10, 11, 18, 13, 14, 15, 24, 17, 24, 19, 30, 21, 22, 23, 36, 30, 26, 36, 42, 29, 30, 31, 48, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 72, 56, 60, 51, 78, 53, 72, 55, 84, 57, 58, 59, 90, 61, 62, 84, 96, 65, 66, 67, 102, 69, 70, 71, 108, 73, 74, 90, 114, 77, 78, 79, 120

Offset

1,2

Comments

Not multiplicative. The least counterexample is 72 = 8*9: a(72) = 108, while a(8) * a(9) = 12 * 12 = 144. - Amiram Eldar, Sep 09 2023

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences related to sigma(n).

Formula

a(n) = Sum_{k=1..n} (1 - ceiling(n/k^2) + floor(n/k^2)) * mu(k)^2 * (n/k), where mu is the Möbius function (A008683). - Wesley Ivan Hurt, Jan 29 2021

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; a[1] = 1; a[n_] := a[n] = Module[{fct = FactorInteger[n]}, If[Max[fct[[;; , 2]]] < 3, Times @@ f @@@ fct, r = Times @@ fct[[;; , 1]]; r*a[n/r]]]; Array[a, 100] (* Amiram Eldar, Sep 09 2023 *)

Prog

(PARI) A336465(n) = if(issquarefree(n), n, my(f=factor(n), r=factorback(f[, 1])); if(issquarefree(n/r), r*sigma(n/r), r*A336465(n/r)));

Crossrefs

Cf. A000203, A007947, A057723.

Sequence in context: A361175 A051377 A369319 * A340774 A345068 A057723

Adjacent sequences: A336462 A336463 A336464 * A336466 A336467 A336468

Keyword

nonn

Author

Antti Karttunen, Jul 31 2020

Extensions

Keyword mult removed by Amiram Eldar, Sep 09 2023

Status

approved