A011264 In the prime factorization of n, increment even powers and decrement odd powers (multiplicative).

1, 1, 1, 8, 1, 1, 1, 4, 27, 1, 1, 8, 1, 1, 1, 32, 1, 27, 1, 8, 1, 1, 1, 4, 125, 1, 9, 8, 1, 1, 1, 16, 1, 1, 1, 216, 1, 1, 1, 4, 1, 1, 1, 8, 27, 1, 1, 32, 343, 125, 1, 8, 1, 9, 1, 4, 1, 1, 1, 8, 1, 1, 27, 128, 1, 1, 1, 8, 1, 1, 1, 108, 1, 1, 125, 8, 1, 1, 1, 32, 243, 1, 1, 8, 1, 1, 1, 4, 1, 27, 1, 8, 1, 1

Offset: 1

Author

Marc LeBrun

Keywords

• easy
• nonn
• mult

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A001221, A004442, A007913, A007947, A011262, A027748, A336643, A356191.

Programs

Haskell

a011264 n = product $ zipWith (^)
                      (a027748_row n) (map a004442 $ a124010_row n)
-- Reinhard Zumkeller, Jun 23 2013

Mathematica

f[n_, k_] := n^(If[EvenQ[k], k + 1, k - 1]); Table[Times @@ f @@@ FactorInteger[n], {n, 94}] (* Jayanta Basu, Aug 14 2013 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^if(f[i,2]%2, f[i,2]-1, f[i,2]+1));} \\ Amiram Eldar, Jan 07 2023

Formula

a(n) = Product_{k=1..A001221(n)} (A027748(n,k)^A004442(A124010(n,k))). - Reinhard Zumkeller, Jun 23 2013

From Amiram Eldar, Jan 07 2023: (Start)

a(n) = n^2/A011262(n).

a(n) = n*A007947(n)/A007913(n)^2.

a(n) = n*A336643(n)/A007913(n).

a(n) = A356191(n)/A007913(n). (End)

Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-3) - 1/p^(2*s-2)). - Amiram Eldar, Sep 21 2023

From Vaclav Kotesovec, May 06 2025: (Start)

Dirichlet g.f.: zeta(2*s-3) * Product_{p prime} (1 + (p-1)*p^(3-2*s) + p^(1-s) - (p-1)*(p^s + p^3)/(p^(2*s) - p^2)).

Sum_{k=1..n} a(k) ~ n^2/4. (End)