A356537 Numbers k whose binary expansion is a substring of the binary expansion of binomial(k,2).

3, 5, 9, 11, 17, 33, 44, 50, 58, 65, 129, 257, 396, 452, 513, 581, 864, 971, 1025, 1139, 1843, 1881, 1914, 2049, 2541, 2676, 2929, 3130, 4097, 4596, 5254, 6621, 7010, 7111, 8193, 10771, 11140, 12655, 16385, 17090, 19135, 19371, 19580, 20985, 27117, 27845, 32769, 35272, 44278, 46779, 56069

Offset

1,1

Comments

All numbers of the form 2^m+1, m>=1, are in the sequence. There are 152 terms below 100 million.

Links

Table of n, a(n) for n=1..51.

Example

9 is a term as 9 = 1001_2 and binomial(9,2) = 9!/(2!7!) = 36 = 100100_2 and "100100" contains "1001" as a substring.

Mathematica

kmax=56100; a={}; For[k=1, k<=kmax, k++, If[StringContainsQ[ToString[FromDigits[IntegerDigits[Binomial[k, 2], 2]]], ToString[FromDigits[IntegerDigits[k, 2]]]], AppendTo[a, k]]]; a (* Stefano Spezia, Aug 11 2022 *)

Prog

(PARI) str(k) = Str(fromdigits(binary(k)));
isok(k) = #strsplit(str(binomial(k, 2)), str(k)) > 1; \\ Michel Marcus, Aug 11 2022
(Python)
from math import comb
def ok(n): return n > 0 and str(bin(n)[2:]) in str(bin(comb(n, 2))[2:])
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Aug 11 2022

Crossrefs

Cf. A000217, A187752, A030190, A351753.

Sequence in context: A120811 A340287 A123069 * A270836 A100456 A059819

Adjacent sequences: A356534 A356535 A356536 * A356538 A356539 A356540

Keyword

nonn,base

Author

Scott R. Shannon, Aug 11 2022

Status

approved