A356353 Numbers k such that A356352(k) <> 1.

0, 3, 7, 12, 15, 31, 48, 51, 56, 60, 63, 127, 192, 195, 204, 207, 240, 243, 252, 255, 448, 455, 504, 511, 768, 771, 780, 783, 816, 819, 828, 831, 960, 963, 972, 975, 992, 1008, 1011, 1020, 1023, 2047, 3072, 3075, 3084, 3087, 3120, 3123, 3132, 3135, 3264, 3267

Offset: 1

Author

Rémy Sigrist, Oct 15 2022

Keywords

• nonn
• base

Comments

Also, numbers whose binary expansions are juxtapositions of constant blocks of size g > 1.

A001196 and A097254 are subsequences.

There are A178472(k) terms with binary length k.

Examples

			The first terms, alongside their binary expansions and A356352(a(n)), are:
  n   a(n)  bin(a(n))   A356352(a(n))
  --  ----  ----------  -------------
   1     0           0              0
   2     3          11              2
   3     7         111              3
   4    12        1100              2
   5    15        1111              4
   6    31       11111              5
   7    48      110000              2
   8    51      110011              2
   9    56      111000              3
  10    60      111100              2
  11    63      111111              6
  12   127     1111111              7
  13   192    11000000              2
  14   195    11000011              2
  15   204    11001100              2
  16   207    11001111              2
		

Links

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

Crossrefs

Cf. A001196, A097254, A178472, A356352.

Programs

PARI

is(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); gcd(r)!=1 }

PARI

See Links section.

Python

from math import gcd
from itertools import groupby
def ok(n):
    if n == 0: return True # by convention of A356352
    return gcd(*(len(list(g)) for k, g in groupby(bin(n)[2:]))) != 1
print([k for k in range(3268) if ok(k)]) # Michael S. Branicky, Oct 15 2022