A367406 The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).

1, 4, 9, 25, 36, 49, 16, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 144, 676, 81, 841, 900, 961, 64, 1089, 1156, 1225, 1369, 1444, 1521, 400, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 324, 3025, 784, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761

Offset

1,2

Comments

Analogous to A355038, with the exponentially odd numbers instead of the square numbers (A000290).

This sequence is a permutation of the square numbers.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A064549(A268335(n)).

a(n) = A268335(n)*A367417(n).

a(n) = A367407(n)^2.

a(n) = A268335(n)^2/A367418(n).

Sum_{k=1..n} a(k) = c * n^3 / 3, where c = (Pi^2/(15*d^3)) * Product_{p prime} (1 - 1/(p^3*(p+1))) = 1.78385074227198915372..., and d = A065463 is the asymptotic density of the exponentially odd numbers.

a(n) = A053143(A268335(n)). - Amiram Eldar, Nov 30 2023

Mathematica

s[n_] := n * Times @@ FactorInteger[n][[;; , 1]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

Prog

(PARI) b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^(f[i, 2]+1), 0)); }
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", "))); }

Crossrefs

Cf. A000290, A007947, A053143, A064549, A065463, A268335, A355038, A367407, A367417, A367418.

Sequence in context: A099998 A360902 A365003 * A030140 A325240 A355058

Adjacent sequences: A367403 A367404 A367405 * A367407 A367408 A367409

Keyword

nonn,easy

Author

Amiram Eldar, Nov 17 2023

Status

approved