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

1, 4, 9, 2, 25, 36, 49, 16, 27, 100, 121, 18, 169, 196, 225, 8, 289, 108, 361, 50, 441, 484, 529, 144, 125, 676, 3, 98, 841, 900, 961, 64, 1089, 1156, 1225, 54, 1369, 1444, 1521, 400, 1681, 1764, 1849, 242, 675, 2116, 2209, 72, 343, 500, 2601, 338, 2809, 12, 3025

Offset: 1

Author

Amiram Eldar, Nov 17 2023

Keywords

• nonn
• easy
• mult

Comments

A permutation of the positive integers. 1 is the only fixed point.

a(n) is a powerful number (A001694) if and only if n is not in A100717.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A000040, A001694, A005117, A007947, A048102, A072873, A100717, A103889, A130508, A342090.

Similar sequences: A011262, A011264.

Programs

Mathematica

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]

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])));}

Formula

a(2^e) = 2^A103889(e).

a(3^e) = 3^A130508(e).

A007947(a(n)) = A007947(n).

a(A051674(n)) = A000040(n).

a(n) is squarefree (A005117) if and only if n is in A048102.

a(A048102(n)) = A007947(A048102(n)).

a(n) == 0 (mod n) if and only if n is not in A342090.

a(n) | n if and only if n is in A072873.

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... .