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

%I A011264 #25 May 06 2025 11:49:05
%S A011264 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,
%T A011264 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,
%U A011264 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
%N A011264 In the prime factorization of n, increment even powers and decrement odd powers (multiplicative).
%H A011264 Reinhard Zumkeller, <a href="/A011264/b011264.txt">Table of n, a(n) for n = 1..10000</a>
%H A011264 Vaclav Kotesovec, <a href="/A011264/a011264.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A011264 a(n) = Product_{k=1..A001221(n)} (A027748(n,k)^A004442(A124010(n,k))). - _Reinhard Zumkeller_, Jun 23 2013
%F A011264 From _Amiram Eldar_, Jan 07 2023: (Start)
%F A011264 a(n) = n^2/A011262(n).
%F A011264 a(n) = n*A007947(n)/A007913(n)^2.
%F A011264 a(n) = n*A336643(n)/A007913(n).
%F A011264 a(n) = A356191(n)/A007913(n). (End)
%F A011264 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
%F A011264 From _Vaclav Kotesovec_, May 06 2025: (Start)
%F A011264 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)).
%F A011264 Sum_{k=1..n} a(k) ~ n^2/4. (End)
%t A011264 f[n_, k_] := n^(If[EvenQ[k], k + 1, k - 1]); Table[Times @@ f @@@ FactorInteger[n], {n, 94}] (* _Jayanta Basu_, Aug 14 2013 *)
%o A011264 (Haskell)
%o A011264 a011264 n = product $ zipWith (^)
%o A011264                       (a027748_row n) (map a004442 $ a124010_row n)
%o A011264 -- _Reinhard Zumkeller_, Jun 23 2013
%o A011264 (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
%Y A011264 Cf. A001221, A004442, A007913, A007947, A011262, A027748, A336643, A356191.
%K A011264 easy,nonn,mult
%O A011264 1,4
%A A011264 _Marc LeBrun_