A357758 Numbers k such that in the binary expansion of k, the Hamming weight of each block differs by at most 1 from every other block of the same length.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 41, 42, 43, 45, 46, 47, 53, 54, 55, 59, 61, 62, 63, 64, 65, 66, 68, 72, 73, 74, 82, 84, 85, 86, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111
Offset: 1
Examples
For k = 42: - the binary expansion of 42 is "101010", - blocks of length 1 have Hamming weight 0 or 1, - blocks of length 2 have Hamming weight 1, - blocks of length 3 have Hamming weight 1 or 2, - blocks of length 4 have Hamming weight 2, - blocks of length 5 have Hamming weight 2 or 3, - so 42 belongs to the sequence. For k = 44: - the binary expansion of 44 is "101100", - blocks of length 2 have Hamming weight 0, 1 or 2, - so 44 does not belong to the sequence.
Links
Programs
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PARI
See Links section.
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Python
def ok(n): b = bin(n)[2:] if "00" in b and "11" in b: return False for l in range(3, len(b)): h = set(b[i:i+l].count("1") for i in range(len(b)-l+1)) if max(h) - min(h) > 1: return False return True print([k for k in range(112) if ok(k)]) # Michael S. Branicky, Oct 12 2022
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