A008473 If n = Product (p_j^k_j) then a(n) = Product (p_j + k_j).

1, 3, 4, 4, 6, 12, 8, 5, 5, 18, 12, 16, 14, 24, 24, 6, 18, 15, 20, 24, 32, 36, 24, 20, 7, 42, 6, 32, 30, 72, 32, 7, 48, 54, 48, 20, 38, 60, 56, 30, 42, 96, 44, 48, 30, 72, 48, 24, 9, 21, 72, 56, 54, 18, 72, 40, 80, 90, 60, 96, 62, 96, 40, 8, 84, 144, 68, 72, 96, 144, 72, 25, 74, 114

Offset: 1

Author

Olivier Gérard

Keywords

• nonn
• mult

Comments

Coincides with sigma (A000203) for squarefree n. - Franklin T. Adams-Watters, Jan 31 2016

Every positive integer except 2 occurs in this sequence, but none occur infinitely often. For m > 4, there are n > m with a(n) = m. This implies that every integer greater than 4 occurs in the iterated sequence infinitely often. For example, 5 <- 8 <- 125 <- 113^12 <- .... - Franklin T. Adams-Watters, Jan 31 2016

Sum of the powerfree parts of the divisors of n. - Wesley Ivan Hurt, Jun 13 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A027748, A124010, A001221, A203908, A000203, A322965, A322966.

Cf. A055231 (powerfree part of n).

Programs

Haskell

a008473 n = product $ zipWith (+) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jul 17 2014

Maple

A008473 := proc(n) local e,j; e := ifactors(n)[2]:
mul (e[j][1]+e[j][2], j=1..nops(e)) end:
seq (A008473(n), n=1..80);
# Peter Luschny, Jan 17 2011

Mathematica

Array[Times @@ Total /@ FactorInteger[ # ] &, 80] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 28 2006 *)

PARI

a(n)=my(f = factor(n)); for (k=1, #f~, f[k, 1] = f[k, 1] + f[k, 2]; f[k, 2] = 1;); factorback(f); \\ Michel Marcus, Jan 31 2016

Formula

Multiplicative with a(p^e) = p+e. - David W. Wilson, Aug 01 2001

a(n) = Product_{k=1..A001221(n)} (A027748(n,k) + A124010(n,k)). - Reinhard Zumkeller, Jul 17 2014

a(n)/A007947(n) = A322965(n)/A322966(n). - David S. Metzler, Jan 01 2019

a(n) = Sum_{d|n} A055231(d). - Wesley Ivan Hurt, Jun 13 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - 2/p^2 + 2/p^4 - 1/p^5) = 0.5342800948... . - Amiram Eldar, Dec 08 2022

Extensions

More terms from Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 28 2006