A336591 Numbers whose exponents in their prime factorization are either 1, 3, or both.

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95

Offset: 1

Author

Amiram Eldar, Jul 26 2020

Keywords

• nonn

Comments

The asymptotic density of this sequence is zeta(6)/(zeta(2) * zeta(3)) * Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5) = 0.68428692418686231814196872579121808347231273672316377728461822629005... (Cohen, 1962).

First differs from A036537 at n = 89. A036537(89) = 128 = 2^7 is not a term of this sequence.

Examples

			1 is a term since it has no exponents, and thus it has no exponent that is not 1 or 3.
2 is a term since 2 = 2^1 has only the exponent 1 in its prime factorization.
24 is a term since 24 = 2^3 * 3^1 has the exponents 1 and 3 in its prime factorization.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, No. 12, Vol. 1 (1962), pp. 77-84.

Crossrefs

Intersection of A046100 and A036537.

Intersection of A046100 and A268335.

A005117 and A062838 are subsequences.

Cf. A068468.

Programs

Mathematica

seqQ[n_] := AllTrue[FactorInteger[n][[;;,2]], MemberQ[{1, 3}, #] &]; Select[Range[100], seqQ]

Python

from itertools import count, islice
from sympy import factorint
def A336591_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e==1 or e==3 for e in factorint(n).values()),count(max(startvalue,1)))
A336591_list = list(islice(A336591_gen(),20)) # Chai Wah Wu, Jun 22 2023