A260619 Arithmetic derivative of hyperfactorial(n).

0, 0, 4, 216, 165888, 604800000, 48372940800000, 43156963184025600000, 1392410948543163924480000000, 668916177911197542484208831692800000, 8199617664717905359483850194944000000000000000, 2401010998878767104110478543683244630474752000000000000000

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..37

Formula

a(n) = A003415(A002109(n)).

a(n) = A002109(n)*A190121(n) (conjectured).

Maple

h:= proc(n) option remember; `if`(n=0, 1, h(n-1)* n^n) end:
a:= proc(n) n^n *`if`(n=0, 0,
      a(n-1)+h(n-1)*n*add(i[2]/i[1], i=ifactors(n)[2]))
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 18 2015

Mathematica

a[n_] := If[n<2, 0, With[{h = Hyperfactorial[n]}, h Sum[{p, e} = pe; e/p, {pe, FactorInteger[h]}]]];
a /@ Range[0, 15] (* Jean-François Alcover, Nov 14 2020 *)

Prog

(Python 3.8+)
from math import prod
from collections import Counter
from sympy import factorint
def A260619(n):
    s = prod(i**i for i in range(2, n+1))
    return sum(s*e//p for p, e in sum(((lambda x: Counter({k:x[k]*m for k in x}))(factorint(m)) for m in range(2, n+1)), start=Counter({2:0})).items()) if n > 1 else 0 # Chai Wah Wu, Jun 12 2022

Crossrefs

Cf. A002109, A003415, A068327.

Sequence in context: A276487 A269283 A055627 * A167888 A371204 A036740

Adjacent sequences: A260616 A260617 A260618 * A260620 A260621 A260622

Keyword

nonn

Author

Matthew Campbell, Sep 17 2015

Extensions

More terms from Alois P. Heinz, Sep 18 2015

Status

approved