A320674 -id:A320674 - cp's OEIS frontend

A029747 Numbers of the form 2^k times 1, 3 or 5.

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 128, 160, 192, 256, 320, 384, 512, 640, 768, 1024, 1280, 1536, 2048, 2560, 3072, 4096, 5120, 6144, 8192, 10240, 12288, 16384, 20480, 24576, 32768, 40960, 49152, 65536, 81920, 98304, 131072, 163840, 196608

Offset: 1

N. J. A. Sloane

Fixed points of the Doudna sequence: A005940(a(n)) = A005941(a(n)) = a(n). - Reinhard Zumkeller, Aug 23 2006
Subsequence of A103969. - R. J. Mathar, Mar 06 2010
Question: Is there a simple proof that A005940(c) = c would never allow an odd composite c as a solution? See also my comments in A163511 and in A335431 concerning similar problems, also A364551 and A364576. - Antti Karttunen, Jul 28 & Aug 11 2023

			128 = 2^7 * 1 is in the sequence as well as 160 = 2^5 * 5. - _David A. Corneth_, Sep 18 2020

David A. Corneth, Table of n, a(n) for n = 1..9963 (terms <= 10^1000)
Index entries for linear recurrences with constant coefficients, signature (0,0,2).

Cf. A122132, A000079, A007283, A020714, A130793.
Cf. A005940, A005941, A163511, A335431, A364499, A364551, A364576.
Subsequence of the following sequences: A103969, A253789, A364541, A364542, A364544, A364546, A364548, A364550, A364560, A364565.
Even terms form a subsequence of A320674.

m = 200000; Select[Union @ Flatten @ Outer[Times, {1, 3, 5}, 2^Range[0, Floor[Log2[m]]]], # < m &] (* Amiram Eldar, Oct 15 2020 *)

is(n) = n>>valuation(n, 2) <= 5 \\ David A. Corneth, Sep 18 2020

def A029747(n):
    if n<3: return n
    a, b = divmod(n,3)
    return 1<Chai Wah Wu, Apr 02 2025

a(n) = if n < 6 then n else 2*a(n-3). - Reinhard Zumkeller, Aug 23 2006
G.f.: (1+x+x^2)^2/(1-2*x^3). - R. J. Mathar, Mar 06 2010
Sum_{n>=1} 1/a(n) = 46/15. - Amiram Eldar, Oct 15 2020

Edited by David A. Corneth and Peter Munn, Sep 18 2020

A320673 Positive integers m with binary expansion (b_1, ..., b_k) (where k = A070939(m)) such that b_i = [m == 0 (mod i)] for i = 1..k (where [] is an Iverson bracket).

Original entry on oeis.org

1, 50, 52, 104, 114, 3460, 12298, 29442, 31368, 856592, 1713184, 54822416, 109578256, 109644832, 219156512, 219289664, 438313024, 438579328, 876626048, 877158656, 1034367516, 1753252096, 1754317312, 112208117792, 113290248736, 224416235584, 226580497472

Offset: 1

Rémy Sigrist, Oct 19 2018

In other words, the binary representation of a term of this sequence encodes the first divisors and nondivisors of this term respectively as ones and zeros.
Is this sequence infinite?
See A320674 and A320675 for similar sequences.

			The first terms, alongside their binary representation and the divisors encoded therein, are:
  n  a(n)   bin(a(n))        First divisors
  -  -----  ---------------  --------------------
  1      1  1                1
  2     50  110010           1, 2, 5
  3     52  110100           1, 2, 4
  4    104  1101000          1, 2, 4
  5    114  1110010          1, 2, 3, 6
  6   3460  110110000100     1, 2, 4, 5, 10
  7  12298  11000000001010   1, 2, 11, 13
  8  29442  111001100000010  1, 2, 3, 6, 7, 14
  9  31368  111101010001000  1, 2, 3, 4, 6, 8, 12

Cf. A070939, A320674, A320675.

is(n) = my (b=binary(n)); b==vector(#b, k, n%k==0)

from itertools import count, islice
def A320673_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:not any(int(b)==bool(n%i) for i,b in enumerate(bin(n)[2:],1)),count(max(startvalue,0)))
A320673_list = list(islice(A320673_gen(),10)) # Chai Wah Wu, Dec 12 2022

cp's OEIS Frontend

A029747 Numbers of the form 2^k times 1, 3 or 5.

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