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A366423 Multiplicative with a(p^e) = p^(e+1-p) if p|e, and p^(e+1) otherwise.

%I A366423 #12 Nov 18 2023 13:24:02
%S A366423 1,4,9,2,25,36,49,16,27,100,121,18,169,196,225,8,289,108,361,50,441,
%T A366423 484,529,144,125,676,3,98,841,900,961,64,1089,1156,1225,54,1369,1444,
%U A366423 1521,400,1681,1764,1849,242,675,2116,2209,72,343,500,2601,338,2809,12,3025
%N A366423 Multiplicative with a(p^e) = p^(e+1-p) if p|e, and p^(e+1) otherwise.
%C A366423 A permutation of the positive integers. 1 is the only fixed point.
%C A366423 a(n) is a powerful number (A001694) if and only if n is not in A100717.
%H A366423 Amiram Eldar, <a href="/A366423/b366423.txt">Table of n, a(n) for n = 1..10000</a>
%H A366423 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A366423 a(2^e) = 2^A103889(e).
%F A366423 a(3^e) = 3^A130508(e).
%F A366423 A007947(a(n)) = A007947(n).
%F A366423 a(A051674(n)) = A000040(n).
%F A366423 a(n) is squarefree (A005117) if and only if n is in A048102.
%F A366423 a(A048102(n)) = A007947(A048102(n)).
%F A366423 a(n) == 0 (mod n) if and only if n is not in A342090.
%F A366423 a(n) | n if and only if n is in A072873.
%F A366423 Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 1/p + 1/(1 + p) - (p-1)/(p^p * (1 + p^p))) = 0.660264348361... .
%t A366423 f[p_, e_] := p^(e + 1 + If[Mod[e, p] == 0, -p, 0]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366423 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + 1 + if(!(f[i,2]%f[i,1]), -f[i,1])));}
%Y A366423 Cf. A000040, A001694, A005117, A007947, A048102, A072873, A100717, A103889, A130508, A342090.
%Y A366423 Similar sequences: A011262, A011264.
%K A366423 nonn,easy,mult
%O A366423 1,2
%A A366423 _Amiram Eldar_, Nov 17 2023