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If n = Product (p_j^k_j) then a(n) = Product (p_j + k_j).
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%I #49 Dec 08 2022 07:36:30

%S 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,

%T 6,32,30,72,32,7,48,54,48,20,38,60,56,30,42,96,44,48,30,72,48,24,9,21,

%U 72,56,54,18,72,40,80,90,60,96,62,96,40,8,84,144,68,72,96,144,72,25,74,114

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

%C Coincides with sigma (A000203) for squarefree n. - _Franklin T. Adams-Watters_, Jan 31 2016

%C 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

%C Sum of the powerfree parts of the divisors of n. - _Wesley Ivan Hurt_, Jun 13 2021

%H Vincenzo Librandi, <a href="/A008473/b008473.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p+e. - _David W. Wilson_, Aug 01 2001

%F a(n) = Product_{k=1..A001221(n)} (A027748(n,k) + A124010(n,k)). - _Reinhard Zumkeller_, Jul 17 2014

%F a(n)/A007947(n) = A322965(n)/A322966(n). - _David S. Metzler_, Jan 01 2019

%F a(n) = Sum_{d|n} A055231(d). - _Wesley Ivan Hurt_, Jun 13 2021

%F 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

%p A008473 := proc(n) local e,j; e := ifactors(n)[2]:

%p mul (e[j][1]+e[j][2], j=1..nops(e)) end:

%p seq (A008473(n), n=1..80);

%p # _Peter Luschny_, Jan 17 2011

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

%o (Haskell)

%o a008473 n = product $ zipWith (+) (a027748_row n) (a124010_row n)

%o -- _Reinhard Zumkeller_, Jul 17 2014

%o (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

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

%Y Cf. A055231 (powerfree part of n).

%K nonn,mult

%O 1,2

%A _Olivier GĂ©rard_

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