A322966 -id:A322966 - cp's OEIS frontend

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

Olivier Gérard

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

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

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

Cf. A055231 (powerfree part of n).

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

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

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

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

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

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

A322965 Numerator of Sum_{d | n} 1/rad(d).

Original entry on oeis.org

1, 3, 4, 2, 6, 2, 8, 5, 5, 9, 12, 8, 14, 12, 8, 3, 18, 5, 20, 12, 32, 18, 24, 10, 7, 21, 2, 16, 30, 12, 32, 7, 16, 27, 48, 10, 38, 30, 56, 3, 42, 16, 44, 24, 2, 36, 48, 4, 9, 21, 24, 28, 54, 3, 72, 20, 80, 45, 60, 16, 62, 48, 40, 4, 84, 24, 68, 36, 32, 72, 72, 25, 74, 57, 28

Offset: 1

David S. Metzler, Dec 31 2018

Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let g(n) = 1/rad(n) and let f(n) = Sum_{d | n} g(d). This is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves. The sequence a(n) lists the numerators of the fractions f(n) in lowest terms.

If p is prime, then a(p^k) = p+k if p does not divide k, 1 + k/p if it does. In particular, a(p^p) = 2. - Robert Israel, Jan 25 2019

			The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3 and a(12) = 8. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3) = 8/3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007947 (radical), A322966 (denominators), A008473 (unreduced numerators, i.e., f(n)*rad(n)), A082695.

Numbers n where f(n) increases to a record: A322447.

rad:= n -> convert(numtheory:-factorset(n),`*`):
f:= proc(n) numer(add(1/rad(d),d=numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Jan 25 2019

Array[Numerator@ DivisorSum[#, 1/Apply[Times, FactorInteger[#][[All, 1]]] &] &, 71] (* Michael De Vlieger, Jan 19 2019 *)

rad(n) = factorback(factor(n)[, 1]); \\ A007947
a(n) = numerator(sumdiv(n, d, 1/rad(d))); \\ Michel Marcus, Jan 10 2019

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A322966(k) = zeta(2)*zeta(3)/zeta(6) (A082695). - Amiram Eldar, Dec 09 2023

More terms from Michel Marcus, Jan 19 2019

cp's OEIS Frontend

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

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