A359593 Multiplicative with a(p^e) = 1 if p divides e, p^e otherwise.

1, 2, 3, 1, 5, 6, 7, 8, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 24, 25, 26, 1, 7, 29, 30, 31, 32, 33, 34, 35, 9, 37, 38, 39, 40, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 56, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71, 72, 73, 74, 75, 19, 77, 78, 79, 5, 81, 82, 83, 21

Offset

1,2

Comments

Each term a(n) is a multiple of both A083346(n) and A327938(n).

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = n / A359594(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p^(p-1)*(p-1)/(p^(2*p)-1)) = 0.4225104173... . - Amiram Eldar, Jan 11 2023

Mathematica

f[p_, e_] := If[Divisible[e, p], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 09 2023 *)

Prog

(PARI) A359593(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^(f[k, 2]*!!(f[k, 2]%f[k, 1]))); };
(Python)
from math import prod
from sympy import factorint
def A359593(n): return prod(p**e for p, e in factorint(n).items() if e%p) # Chai Wah Wu, Jan 10 2023

Crossrefs

Cf. A072873 (positions of 1's), A359594.

Cf. also A083346, A327938, A342014, A359552.

Sequence in context: A350389 A368886 A182659 * A358881 A197701 A292770

Adjacent sequences: A359590 A359591 A359592 * A359594 A359595 A359596

Keyword

nonn,mult

Author

Antti Karttunen, Jan 09 2023

Status

approved