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A011262 In the prime factorization of n, increment odd powers and decrement even powers (multiplicative and self-inverse).

%I A011262 #28 Dec 30 2024 03:57:35
%S A011262 1,4,9,2,25,36,49,16,3,100,121,18,169,196,225,8,289,12,361,50,441,484,
%T A011262 529,144,5,676,81,98,841,900,961,64,1089,1156,1225,6,1369,1444,1521,
%U A011262 400,1681,1764,1849,242,75,2116,2209,72,7,20,2601,338,2809,324,3025,784,3249
%N A011262 In the prime factorization of n, increment odd powers and decrement even powers (multiplicative and self-inverse).
%H A011262 Paul Tek, <a href="/A011262/b011262.txt">Table of n, a(n) for n = 1..10000</a>
%H A011262 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A011262 Multiplicative with f(p^k) = p^(k-1) if k even, p^(k+1) if k odd.
%F A011262 a(n) = Product_{k = 1..A001221(n)} A027748(n,k) ^ A103889(A124010(n,k)). - _Reinhard Zumkeller_, Jun 23 2013
%F A011262 Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^5 + p^4 - p + 1)/(p^5 + p^4 + p^3 + p^2)) = 0.21311151701724196530... . - _Amiram Eldar_, Oct 13 2022
%t A011262 f[n_, k_] := n^(If[EvenQ[k], k - 1, k + 1]); Table[Times @@ f @@@ FactorInteger[n], {n, 57}] (* _Jayanta Basu_, Aug 14 2013 *)
%o A011262 (PARI) a(n)=my(f=factor(n));return(prod(i=1,#f[,1],f[i,1]^(f[i,2]-(-1)^f[i,2]))) \\ _Paul Tek_, Jun 01 2013
%o A011262 (Haskell)
%o A011262 a011262 n = product $ zipWith (^)
%o A011262                       (a027748_row n) (map a103889 $ a124010_row n)
%o A011262 -- _Reinhard Zumkeller_, Jun 23 2013
%Y A011262 Cf. A001221, A011264, A027748, A103889, A124010.
%K A011262 nonn,easy,mult
%O A011262 1,2
%A A011262 _Marc LeBrun_