A339746 Positive integers of the form 2^i*3^j*k, gcd(k,6)=1, and i == j (mod 3).

1, 5, 6, 7, 8, 11, 13, 17, 19, 23, 25, 27, 29, 30, 31, 35, 36, 37, 40, 41, 42, 43, 47, 48, 49, 53, 55, 56, 59, 61, 64, 65, 66, 67, 71, 73, 77, 78, 79, 83, 85, 88, 89, 91, 95, 97, 101, 102, 103, 104, 107, 109, 113, 114, 115, 119, 121, 125, 127, 131, 133, 135

Offset

1,2

Comments

From Peter Munn, Mar 16 2021: (Start)

The positive integers in the multiplicative subgroup of the positive rationals generated by 8, 6, and A215848 (primes greater than 3).

This subgroup, denoted H, has two cosets: 2H = (1/3)H and 3H = (1/2)H. It follows that the sequence is one part of a 3-part partition of the positive integers with the property that each part's terms are half the even terms of one of the other parts and also one third of the multiples of 3 in the remaining part.

(End)

Positions of multiples of 3 in A276085. Because A276085 is completely additive, this is closed under multiplication: if m and n are in the sequence then so is m*n. - Antti Karttunen, May 27 2024

The coset sequences mentioned in Peter Munn's comment above are A373261 and A373262. - Antti Karttunen, Jun 04 2024

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) ~ (91/43)*n.

Maple

N:= 1000: # for terms <= N
R:= {}:
for k1 from 0 to floor(N/6) do
  for k0 in [1, 5] do
    k:= k0 + 6*k1;
    for j from 0 while 3^j*k <= N do
      for i from (j mod 3) by 3 do
        x:= 2^i * 3^j * k;
        if x > N then break fi;
        R:= R union {x}
od od od od:
sort(convert(R, list)); # Robert Israel, Apr 08 2021

Mathematica

Select[Range[130], Mod[IntegerExponent[#, 2] - IntegerExponent[#, 3], 3] == 0 &]

Prog

(PARI) isA339746 = A372573; \\ Antti Karttunen, Jun 04 2024
(Python)
from sympy import factorint
def ok(n):
  f = factorint(n, limit=4)
  i, j = 0 if 2 not in f else f[2], 0 if 3 not in f else f[3]
  return (i-j)%3 == 0
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(200)) # Michael S. Branicky, Mar 26 2021

Crossrefs

Sequences of positive integers in a multiplicative subgroup of positive rationals generated by a set S and A215848: S={}: A007310, S={6}: A064615, S={3,4}: A003159, S={2,9}: A007417, S={4,6}: A036668, S={3,8}: A191257, S={4,9}: A339690, S={6,8}: this sequence.

Positions of 0's in A373153, positions of multiples of 3 in A276085 and in A372576.

Cf. A372573 (characteristic function), A373261, A373262.

Subsequences: A064615, A373144, A373373, A373484, A373837, A374042, A374044, A374120.

Sequences giving positions of multiples of k in A276085, for k=2, 3, 4, 5, 8: A003159, this sequence, A369002, A373140, A373138.

Cf. also A332820, A373992.

Sequence in context: A263093 A115839 A200259 * A262513 A130206 A178502

Adjacent sequences: A339743 A339744 A339745 * A339747 A339748 A339749

Keyword

nonn,changed

Author

Griffin N. Macris, Dec 15 2020

Status

approved