A381934 -id:A381934 - cp's OEIS frontend

A381935 For any n > 0, a(n) is the least nontrivial multiple of n whose binary expansion has the same number of nonleading zeros as that of n; a(0) = 0.

0, 3, 6, 15, 12, 30, 30, 63, 24, 45, 60, 55, 60, 247, 126, 255, 48, 85, 90, 57, 120, 189, 110, 253, 120, 175, 494, 8127, 252, 1015, 510, 1023, 96, 165, 170, 105, 180, 185, 114, 117, 240, 205, 378, 215, 220, 765, 506, 14335, 240, 343, 350, 765, 988, 159, 16254

Offset: 0

Rémy Sigrist, Mar 10 2025

This sequence is well defined: for any n > 0: write n as m * 2^k for some odd number m; m divides some Mersenne number M > m (see A352217); M * 2^(k + A023416(m)) is a multiple of n with the desired properties, hence a(n) exists.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     3       1         11
   2     6      10        110
   3    15      11       1111
   4    12     100       1100
   5    30     101      11110
   6    30     110      11110
   7    63     111     111111
   8    24    1000      11000
   9    45    1001     101101
  10    60    1010     111100
  11    55    1011     110111
  12    60    1100     111100
  13   247    1101   11110111
  14   126    1110    1111110
  15   255    1111   11111111
  16    48   10000     110000

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program

Cf. A023416, A161399, A352217, A381934.

PARI
```
\\ See Links section.
```

a(n) = n * A381934(n).

A381754 Numbers k such that k and 3*k have the same number of zeros in their binary expansions.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 19, 32, 35, 38, 39, 53, 64, 67, 70, 71, 76, 78, 79, 101, 105, 106, 117, 128, 131, 134, 135, 140, 142, 143, 152, 156, 158, 159, 197, 201, 202, 209, 210, 212, 229, 233, 234, 245, 256, 259, 262, 263, 268, 270, 271, 280, 284, 286, 287, 301, 304

Offset: 1

Barak Manos, Mar 06 2025

If n is in the sequence, so is 2n, hence the sequence is infinite. - Charles R Greathouse IV, Mar 06 2025

This sequence corresponds to the numbers m such that A381934(m) <= 3. - Rémy Sigrist, Mar 12 2025

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

Cf. A023416, A077459, A381934.

filter:= proc(n) numboccur(0,convert(n,base,2)) = numboccur(0,convert(3*n,base,2)) end proc:
select(filter, [$0..400]); # Robert Israel, Apr 07 2025

Select[Range[0, 320], Equal @@ DigitCount[{#, 3*#}, 2, 0] &] (* Amiram Eldar, Mar 06 2025 *)

nz(n) = if(n == 0, 1, 1+logint(n, 2) - hammingweight(n))
is(n)=nz(n)==nz(3*n) \\ Charles R Greathouse IV, Mar 06 2025

def ok(n): return bin(n).count('0') == bin(n * 3).count('0')

cp's OEIS Frontend

A381935 For any n > 0, a(n) is the least nontrivial multiple of n whose binary expansion has the same number of nonleading zeros as that of n; a(0) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula