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

Author

Rémy Sigrist, Mar 10 2025

Keywords

• nonn
• base

Comments

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.

Examples

			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
		

Links

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

Crossrefs

Cf. A023416, A161399, A352217, A381934.

Programs

PARI

\\ See Links section.

Formula

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