A182120 -id:A182120 - cp's OEIS frontend

A366762 Numbers whose canonical prime factorization contains only exponents which are congruent to 1 modulo 3.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Amiram Eldar, Oct 21 2023

First differs from A274034 at n = 42, and from A197680 and A361177 at n = 84.

The asymptotic density of this sequence is zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A002117 * A330523 = A253905 * A065465 = 0.644177671086029533405... .

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

Similar sequences with exponents of a given form: A000290 (2*k), A268335 (2*k+1), A000578 (3*k), A182120 (3*k+2).
Cf. A002117, A065465, A253905, A330523.
Cf. A197680, A274034, A361177, A366761.

q[n_] := AllTrue[FactorInteger[n][[;; , 2]], Mod[#, 3] == 1 &]; Select[Range[120], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2]%3 != 1, return(0))); 1;}

Sum_{n>=1} 1/a(n)^s = zeta(3*s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)), for s > 1.

A366761 Numbers that have at least one exponent in their canonical prime factorization that is divisible by 3.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 64, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 192, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 320, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 432, 440, 448, 456, 459, 472, 488, 500, 504, 512, 513, 520

Offset: 1

Amiram Eldar, Oct 21 2023

Each term has a unique representation of as product of two numbers: one is a cube (A000578) and the second is a number that is not in this sequence.
The asymptotic density of this sequence is 1 - zeta(3) * Product_{p prime} (1 - 2/p^3 + 1/p^4) = 0.10483363599014046584... .
From Amiram Eldar, Jan 22 2024: (Start)
The complement of this sequence is the sequence of numbers called "unitarily 3-free", or "3-skew", by Cohen (1961).
He proved that the asymptotic density of unitarily k-free, i.e., numbers whose prime factorization contain no exponent that is divisible by k, is zeta(k) * Product_{p prime} (1 - 2/p^k + 1/p^(k+1)) (see p. 228, eq. 3.18). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.

A000578 is a subsequence.

Cf. A002117, A182120, A366762.

q[n_] := ! AllTrue[FactorInteger[n][[;; , 2]], ! Divisible[#, 3] &]; Select[Range[500], q]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(!(f[i, 2]%3), return(1))); 0;}

Sum_{n>=1} 1/a(n)^s = zeta(s) * (1 - zeta(3*s) * Product_{p prime} (1 - 2/p^(3*s) + 1/p^(4*s))), for s > 1.

cp's OEIS Frontend

A366762 Numbers whose canonical prime factorization contains only exponents which are congruent to 1 modulo 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula