A390962 The values of A390868(k) when k runs over the integers that are not exponentially odd.

1, 2, 3, 2, 4, 5, 4, 7, 15, 11, 10, 6, 6, 8, 13, 15, 14, 4, 17, 16, 12, 19, 10, 6, 21, 20, 23, 12, 22, 37, 27, 14, 29, 26, 10, 31, 28, 33, 35, 46, 18, 37, 24, 34, 39, 12, 41, 12, 38, 43, 28, 22, 75, 47, 12, 67, 44, 32, 51, 46, 26, 53, 55, 78, 57, 52, 59, 30, 20

Offset

1,2

Comments

The nonzero terms of A390868.

For numbers k that are exponentially odd (A268335), A390868(k) = 0.

Links

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

Formula

a(n) = A390868(A072587(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1 - zeta(3) * Product_{p prime} (1 - 2/p^3 + 2/p^4 - 1/p^5)) / (1 - d) = 0.20497327470585609638..., where d = Product_{p prime} (1 - 1/(p*(p+1))) (A065463) is the asymptotic density of the exponentially odd numbers.

Mathematica

f[p_, e_] := p^e - If[EvenQ[e], (p - 1)*p^(e/2 - 1), 0]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], OddQ], Nothing, n - Times @@ f @@@ fct]]; Array[s, 300]

Prog

(PARI) list(kmax) = {my(f); for(k = 1, kmax, f = factor(k); if(vecprod(apply(x -> x%2, f[, 2])) == 0, print1(k - prod(i = 1, #f~, f[i, 1]^f[i, 2] - if(!(f[i, 2] % 2), (f[i, 1]-1)*f[i, 1]^(f[i, 2]/2 - 1))), ", "))); }

Crossrefs

Cf. A065463, A072587, A268335, A390868.

Sequence in context: A359369 A090321 A241255 * A174625 A178853 A344646

Adjacent sequences: A390959 A390960 A390961 * A390963 A390964 A390965

Keyword

nonn,easy

Author

Amiram Eldar, Nov 25 2025

Status

approved