A343430 Part of n composed of prime factors of the form 3k-1.

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 11, 4, 1, 2, 5, 16, 17, 2, 1, 20, 1, 22, 23, 8, 25, 2, 1, 4, 29, 10, 1, 32, 11, 34, 5, 4, 1, 2, 1, 40, 41, 2, 1, 44, 5, 46, 47, 16, 1, 50, 17, 4, 53, 2, 55, 8, 1, 58, 59, 20, 1, 2, 1, 64, 5, 22, 1, 68, 23, 10, 71, 8, 1, 2, 25, 4, 11, 2, 1, 80, 1, 82, 83, 4, 85

Offset: 1

Author

Peter Munn, Jun 08 2021

Keywords

• nonn
• easy
• mult

Comments

Largest term of A004612 that divides n.

Modulo 6, the prime numbers are partitioned into 4 nonempty sets: {2}, {3}, primes of the form 6k-1 (A007528) and primes of the form 6k+1 (A002476). The modulo 3 partition is nearly the same, but unites the only even prime, 2, with primes of the form 6k-1 in the set of primes we use here.

A positive integer m is a Loeschian number (a term of A003136) if and only if a(A007913(m)) = 1, that is the squarefree part of m has no prime factors of the form 3k-1.

Examples

			n = 60 has prime factorization 60 = 2 * 2 * 3 * 5. Factors 2 = 3*1 - 1 and 5 = 3*2 - 1 have form 3k-1, whereas 3 does not (having form 3k). Multiplying the factors of form 3k-1, we get 2 * 2 * 5 = 20. So a(60) = 20.
		

Crossrefs

Equivalent sequences for prime factors of other forms: A006519 (2 only), A000265 (2k+1), A038500 (3 only), A038502 (3k+/-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A343431 (6k-1).

Range of terms: A004612 (closure under multiplication of A003627).

Cf. A002476, A007528, squarefree part (A007913) of terms of A003136.

First 28 terms are the same as A247503.

Programs

Mathematica

f[p_, e_] := If[Mod[p, 3] == 2, p^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 11 2021 *)

PARI

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 3, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

Python

from math import prod
from sympy import factorint
def A343430(n): return prod(p**e for p, e in factorint(n).items() if p%3==2) # Chai Wah Wu, Dec 23 2022

Formula

Completely multiplicative with a(p) = p if p is of the form 3k-1, otherwise a(p) = 1.

For k >= 1, a(n) = a(k*n) / gcd(k, a(k*n)).

a(n) = A006519(n) * A343431(n).

a(n) = (n / A038500(n)) / A248909(n) = A038502(n) / A248909(n).

A006519(a(n)) = a(A006519(n)) = A006519(n).

A343431(a(n)) = a(A343431(n)) = A343431(n).

A038500(a(n)) = a(A038500(n)) = 1.

A248909(a(n)) = a(A248909(n)) = 1.