A360388 Positive integers with binary expansion (b(1), ..., b(m)) such that Sum_{i = 1..m-k} b(i)*b(i+k) is odd for all k = 0..m-1.

1, 11, 13, 2787, 3189, 36783, 37063, 43331, 47803, 49813, 56669, 58121, 62961, 9205487, 16215601, 23070091, 23248907, 27264653, 27475981, 43469906355, 55167946629, 75985591407, 80056245671, 81489328999, 83389490039, 87235136243, 88437433811, 90400346819

Offset: 1

Author

Rémy Sigrist, Feb 05 2023

Keywords

• nonn
• base

Comments

Leading zeros in binary expansions are ignored.

All terms are odd and odious (A092246).

This sequence is infinite since we can, from a given term, build another larger term (see Guy reference).

See A053006 for the distinct binary lengths.

If m is a term, then A030101(m) is also a term.

Examples

			For n = 11:
- the binary expansion of 11 is b = (1,1,0,1),
- b(1)*b(1) + b(2)*b(2) + b(3)*b(3) + b(4)*b(4) = 1 + 1 + 0 + 1 = 3 is odd,
- b(1)*b(2) + b(2)*b(3) + b(3)*b(4) = 1 + 0 + 0 = 1 is odd,
- b(1)*b(3) + b(2)*b(4) = 0 + 1 = 1 is odd,
- b(1)*b(4) = 1 is odd,
- so 11 belongs to the sequence.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, E38.

Links

• Rémy Sigrist, PARI program
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A030101, A053006, A092246.

Programs

PARI

See Links section.

Python

from itertools import count, islice
from functools import reduce
from operator import ixor
def A360388_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        b = tuple(int(d) for d in bin(n)[2:])
        m = len(b)
        if all(reduce(ixor, (b[i]&b[i+k] for i in range(m-k))) for k in range(m)):
            yield n
A360388_list = list(islice(A360388_gen(),10)) # Chai Wah Wu, Feb 07 2023