A359550 Characteristic function of A048103; From n >= 1, a(n) is multiplicative with a(p^e) = 1 if p > e, otherwise 0. a(0) = 0 by convention.

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1

Offset

0

Links

Antti Karttunen, Table of n, a(n) for n = 0..100000

Index entries for characteristic functions.

Formula

Multiplicative with a(p^e) = [e < p], where [ ] is the Iverson bracket.

a(n) = [A129251(n) == 0] = [A327936(n) == 1] = [A342007(n) == 1].

a(n) = 1 - A342023(n).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 - 1/p^p) = 0.7219902344... . - Amiram Eldar, Jan 07 2023

Example

a(0) = 0 because 0 is a multiple of all numbers, including also all numbers of the form p^p, with p prime.

Mathematica

f[p_, e_] := If[e < p, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 06 2023 *)

Prog

(PARI) A359550(n) = if(!n, n, my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])));
(PARI) A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); }; \\ Antti Karttunen, Nov 13 2024
(Python)
from sympy import factorint
def A359550(n): return int(all(map(lambda d:d[0]>d[1], factorint(n).items()))) # Chai Wah Wu, Jan 06 2023

Crossrefs

Characteristic function of A048103.

Cf. A129251, A327936, A328308, A342007, A342023 (one's complement), A359546, A359551 (Dirichlet inverse), A368915 [= a(A003415(n))], A377868 [= a(A276085(n))], A377982 (partial sums).

Sequence in context: A284939 A188260 A341625 * A166486 A046978 A075553

Adjacent sequences: A359547 A359548 A359549 * A359551 A359552 A359553

Keyword

nonn,mult

Author

Antti Karttunen, Jan 06 2023

Extensions

Term a(0)=0 prepended and name edited accordingly - Antti Karttunen, Nov 13 2024

Status

approved