A342023 a(n) = 1 if there is a prime p such that p^p divides n, otherwise 0.

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

Offset

1

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions.

Formula

a(n) = min(1, A129251(n)) = [A129251(n) > 0], where [ ] is the Iverson bracket.

a(n) = [A342007(n) > 1] = [A327936(n) > 1].

For all n >= 1,

a(n) <= A107078(n), i.e., a(n) = 1 => A107078(n) = 1.

a(n) <= A342024(n), i.e., a(n) = 1 => A342024(n) = 1.

a(n) <= A341999(n), i.e., a(n) = 1 => A341999(n) = 1.

A342004

For all n >= 2, a(A003415(n)) = A341996(n).

For all n >= 0, a(A276086(n)) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 - Product_{p prime} (1 - 1/p^p) = 0.2780097655... . - Amiram Eldar, Jul 24 2022

a(n) = 1 - A359550(n) = A341999(n) - A359546(n). - Antti Karttunen, Jan 06 2023

Mathematica

Array[Function[{D, q}, Boole[Total@ Table[Count[D, _?(IntegerExponent[#, p] == p &)], {p, Prime@ Range@ q}] > 0]] @@ {Divisors[#], PrimePi@ Floor[Sqrt[#]]} &, 120] (* Michael De Vlieger, Mar 11 2021 *)

Prog

(PARI) A342023(n) = if(1==n, 0, my(f = factor(n)); for(k=1, #f~, if(f[k, 2]>=f[k, 1], return(1))); (0));
(Python)
from sympy import factorint
def A342023(n):
    f = factorint(n)
    for p in f:
        if p <= f[p]:
            return 1
    return 0 # Chai Wah Wu, Mar 09 2021

Crossrefs

Characteristic function of A100716.

Cf. A003415, A008966, A048103 (positions of zeros), A107078, A276086, A341996, A341999, A327936, A341999, A342004, A342007, A359546, A359550 (one's complement).

Differs from A129251 and A276077 for the first time at n=108, as here a(108) = 1.

Differs from A342024 for the first time at n=625, where a(625)=0, while A342024(625)=1.

Sequence in context: A287372 A188221 A011765 * A342024 A285464 A331282

Adjacent sequences: A342020 A342021 A342022 * A342024 A342025 A342026

Keyword

nonn

Author

Antti Karttunen, Mar 09 2021

Status

approved