A327190 For any n > 0: consider the different ways to split the binary representation of 2*n+1 into two nonempty parts, say with value x and y; a(n) is the least possible value of x * y.

1, 1, 3, 1, 3, 3, 7, 1, 3, 5, 7, 3, 9, 7, 15, 1, 3, 5, 7, 5, 11, 11, 15, 3, 9, 13, 21, 7, 21, 15, 31, 1, 3, 5, 7, 9, 11, 13, 15, 5, 15, 21, 23, 11, 27, 23, 31, 3, 9, 15, 21, 13, 33, 27, 45, 7, 21, 29, 49, 15, 45, 31, 63, 1, 3, 5, 7, 9, 11, 13, 15, 9, 19, 21

Offset: 1

Author

Rémy Sigrist, Aug 25 2019

Keywords

• nonn
• base

Comments

All terms are odd.

Examples

			For n=42:
- the binary representation of 85 is "1010101",
- there are 6 ways to split it:
   - "1" and "010101": x=1 and y=21: 1 * 21 = 21,
   - "10" and "10101": x=2 and y=21: 2 * 21 = 42,
   - "101" and "0101": x=5 and y=5: 5 * 5 = 25,
   - "1010" and "101": x=10 and y=5: 10 * 5 = 50,
   - "10101" and "01": x=21 and y=1: 21 * 1 = 21,
   - "101010" and "1": x=42 and y=1: 42 * 1 = 42,
- hence a(42) = 21.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..8192

Crossrefs

See A327186 for other variants.

Cf. A000225.

Programs

PARI

a(n) = my (v=oo, b=binary(2*n+1)); for (w=1, #b-1, v=min(v, (fromdigits(b[1..w],2) * fromdigits(b[w+1..#b],2)))); v

Formula

a(n) = 1 iff n is a power of 2.

a(n) = n iff n is a positive Mersenne number (A000225). - Bernard Schott, Aug 26 2019