A367417 The squarefree kernels of the exponentially odd numbers: a(n) = A007947(A268335(n)).

1, 2, 3, 5, 6, 7, 2, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 6, 26, 3, 29, 30, 31, 2, 33, 34, 35, 37, 38, 39, 10, 41, 42, 43, 46, 47, 51, 53, 6, 55, 14, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 22, 89, 91, 93, 94, 95

Offset: 1

Author

Amiram Eldar, Nov 17 2023

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A007947, A268335, A367406.

Cf. A013662, A065463.

Programs

Mathematica

s[n_] := Times @@ FactorInteger[n][[;; , 1]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

PARI

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

Formula

a(n) = A367406(n)/A268335(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (zeta(4)/d^2) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 1.25661974314007532208..., and d = A065463 is the asymptotic density of the exponentially odd numbers.