A361943 a(n) is the least multiple of n whose binary expansion is an abelian square (A272653).

3, 10, 3, 36, 10, 36, 63, 136, 9, 10, 33, 36, 130, 154, 15, 528, 34, 36, 190, 520, 63, 132, 46, 528, 150, 130, 54, 588, 725, 150, 1023, 2080, 33, 34, 630, 36, 222, 190, 156, 520, 615, 588, 43, 132, 45, 46, 235, 528, 147, 150, 51, 156, 53, 54, 165, 2296, 513

Offset: 1

Author

Rémy Sigrist, Mar 31 2023

Keywords

• nonn
• base

Comments

This sequence is well defined as for any n > 0, A020330(n) is a multiple of n and its binary expansion is an abelian square.

Examples

			The first terms, alongside their binary expansion, are:
  n   a(n)  bin(a(n))
  --  ----  ----------
   1     3          11
   2    10        1010
   3     3          11
   4    36      100100
   5    10        1010
   6    36      100100
   7    63      111111
   8   136    10001000
   9     9        1001
  10    10        1010
  11    33      100001
  12    36      100100
  13   130    10000010
  14   154    10011010
  15    15        1111
  16   528  1000010000
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A020330, A272653, A361944.

Programs

PARI

a(n) = { forstep (m = n, oo, n, my (w = #binary(m)); if (w%2==0 && hammingweight(m)==2*hammingweight(m % (2^(w/2))), return (m))) }

Python

from itertools import count
def a(n): return next(m for m in count(n, n) if not (w:=m.bit_length())&1 and m.bit_count() == ((m>>(w>>1)).bit_count())<<1)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 31 2023 after Rémy Sigrist

Formula

a(n) = A361944(n) * n.

a(n) <= A020330(n).

a(n) >= n with equality iff n belongs to A272653.