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A175356 Those positive integers n where, when written in binary, there are exactly k number of runs (of either 0's or 1's) each of exactly k length, for all k where 1<=k<=m, for some positive integer m. 2

%I #26 Aug 05 2021 13:19:12

%S 1,19,25,27,8984,8988,9016,9100,9112,9116,9784,10008,10012,10040,

%T 12568,12572,12600,12680,12686,12728,12740,12742,12744,12750,12760,

%U 12764,12856,13192,13198,13240,13880,14104,14108,14136,14476,14488,14492,14532,14534,14536

%N Those positive integers n where, when written in binary, there are exactly k number of runs (of either 0's or 1's) each of exactly k length, for all k where 1<=k<=m, for some positive integer m.

%C A "run" of 0's is not immediately bounded by any 0's, and a "run" of 1's is not immediately bounded by any 1's.

%C There are exactly (m*(m+1)/2)! / Product_{k=1 to m} k! numbers in the sequence each of m^3/3 + m^2/2 + m/6 binary digits, for all m >= 1, and none of any other number of binary digits.

%C A005811(a(n)) is triangular, i.e., in A000217. - _Michael S. Branicky_, Jan 19 2021

%H Rémy Sigrist, <a href="/A175356/b175356.txt">Table of n, a(n) for n = 1..12664</a>

%H Rémy Sigrist, <a href="/A175356/a175356.gp.txt">PARI program for A175356</a>

%e 9016 in binary is 10001100111000. There is exactly one run of one binary digit, two runs of two binary digits, and three runs of three binary digits. (Note that it doesn't matter if the runs are of 0's or of 1's.) So, 9016 is in the sequence.

%o (PARI) \\ See Links section.

%o (Python)

%o from itertools import groupby

%o def ok(n):

%o runlens = [len(list(g)) for k, g in groupby(bin(n)[2:])]

%o return all(runlens.count(k) == k for k in range(1, max(runlens)+1))

%o def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]

%o print(aupto(14536)) # _Michael S. Branicky_, Jan 19 2021

%Y Cf. A000330, A022915, A175061, A175357.

%Y Cf. A005811, A000217.

%K base,nonn

%O 1,2

%A _Leroy Quet_, Apr 22 2010

%E More terms from _Rémy Sigrist_, Feb 06 2019

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Last modified March 28 16:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)