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A008473
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If n = Product (p_j^k_j) then a(n) = Product (p_j + k_j).
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8
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MAPLE
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A008473 := proc(n) local e, j; e := ifactors(n)[2]:
mul (e[j][1]+e[j][2], j=1..nops(e)) end:
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MATHEMATICA
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Array[Times @@ Total /@ FactorInteger[ # ] &, 80] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 28 2006 *)
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PROG
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(Haskell)
a008473 n = product $ zipWith (+) (a027748_row n) (a124010_row n)
(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
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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More terms from Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 28 2006
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STATUS
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approved
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