A339746 - OEIS

%I #53 Nov 11 2024 12:54:37

%S 1,5,6,7,8,11,13,17,19,23,25,27,29,30,31,35,36,37,40,41,42,43,47,48,

%T 49,53,55,56,59,61,64,65,66,67,71,73,77,78,79,83,85,88,89,91,95,97,

%U 101,102,103,104,107,109,113,114,115,119,121,125,127,131,133,135

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

%C From _Peter Munn_, Mar 16 2021: (Start)

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

%C 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.

%C (End)

%C 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

%C The coset sequences mentioned in _Peter Munn_'s comment above are A373261 and A373262. - _Antti Karttunen_, Jun 04 2024

%H Robert Israel, <a href="/A339746/b339746.txt">Table of n, a(n) for n = 1..10000</a>

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

%p N:= 1000: # for terms <= N

%p R:= {}:

%p for k1 from 0 to floor(N/6) do

%p   for k0 in [1,5] do

%p     k:= k0 + 6*k1;

%p     for j from 0 while 3^j*k <= N do

%p       for i from (j mod 3) by 3 do

%p         x:= 2^i * 3^j * k;

%p         if x > N then break fi;

%p         R:= R union {x}

%p od od od od:

%p sort(convert(R,list)); # _Robert Israel_, Apr 08 2021

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

%o (PARI) isA339746 = A372573; \\ _Antti Karttunen_, Jun 04 2024

%o (Python)

%o from sympy import factorint

%o def ok(n):

%o   f = factorint(n, limit=4)

%o   i, j = 0 if 2 not in f else f[2], 0 if 3 not in f else f[3]

%o   return (i-j)%3 == 0

%o def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]

%o print(aupto(200)) # _Michael S. Branicky_, Mar 26 2021

%Y 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.

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

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

%Y Subsequences: A064615, A373144, A373373, A373484, A373837, A374042, A374044, A374120, A377872.

%Y Sequences giving positions of multiples of k in A276085, for k=2, 3, 4, 5, 8, 27: A003159, this sequence, A369002, A373140, A373138, A377872.

%Y Cf. also A332820, A373992.

%K nonn

%O 1,2

%A _Griffin N. Macris_, Dec 15 2020