A384054 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is an exponentially odd number.

1, 2, 3, 3, 5, 6, 7, 8, 8, 10, 11, 9, 13, 14, 15, 15, 17, 16, 19, 15, 21, 22, 23, 24, 24, 26, 27, 21, 29, 30, 31, 32, 33, 34, 35, 24, 37, 38, 39, 40, 41, 42, 43, 33, 40, 46, 47, 45, 48, 48, 51, 39, 53, 54, 55, 56, 57, 58, 59, 45, 61, 62, 56, 63, 65, 66, 67, 51

Offset

1,2

Links

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

Formula

Multiplicative with a(p^e) = p^e if e is odd, and p^e-1 if e is even.

a(n) = n * A047994(n) / A384052(n).

a(n) = A047994(A350388(n)) * A350389(n).

Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 - 2/p^(2*s) + 1/p^(3*s-1)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 0.95692470821076622881... .

Mathematica

f[p_, e_] := p^e - If[OddQ[e], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] - if(f[i, 2]%2, 0, 1)); }
(Python)
from math import prod
from sympy import factorint
def A384054(n): return prod(p**e-(e&1^1) for p, e in factorint(n).items()) # Chai Wah Wu, May 21 2025

Crossrefs

Unitary analog of A384041.

Cf. A013662, A268335, A350388, A350389, A384046, A384047.

The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), this sequence (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

Sequence in context: A384048 A390863 A390867 * A334819 A338375 A220838

Adjacent sequences: A384051 A384052 A384053 * A384055 A384056 A384057

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 18 2025

Status

approved