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

1,2

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.

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.

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Intersection of A046100 and A036537.

Intersection of A046100 and A268335.

A005117 and A062838 are subsequences.

Cf. A068468.

Sequence in context: A162644 A268335 A002035 * A036537 A072510 A084116

Adjacent sequences: A336588 A336589 A336590 * A336592 A336593 A336594

Keyword

nonn

Author

Amiram Eldar, Jul 26 2020

Status

approved