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A381720 Integers whose Hamming weight is a cube.

%I A381720 #40 May 19 2025 16:32:40
%S A381720 0,1,2,4,8,16,32,64,128,255,256,383,447,479,495,503,507,509,510,512,
%T A381720 639,703,735,751,759,763,765,766,831,863,879,887,891,893,894,927,943,
%U A381720 951,955,957,958,975,983,987,989,990,999,1003,1005,1006,1011,1013,1014
%N A381720 Integers whose Hamming weight is a cube.
%C A381720 The plot of terms of the form k - 2^floor(log_2(k)) shows quasi-periodic structures on the intervals [2^i, 2^(i+1)]. In general, all sequences of the form "Integers whose Hamming weight is f(x)", where f(x) is an integer valued function, are quasi-periodic on intervals [2^i, 2^(i+1)].
%C A381720 The powers of 2 (A000079) are terms.
%C A381720 A023690 is a subsequence.
%H A381720 Robert Israel, <a href="/A381720/b381720.txt">Table of n, a(n) for n = 1..10000</a>
%e A381720 For k = 255: A000120(255) = 8 = 2^3 is a cube, thus 255 is a term.
%p A381720 filter:= proc(n) type(surd(convert(convert(n,base,2),`+`),3),integer) end proc:
%p A381720 select(filter, [$0..2000]); # _Robert Israel_, May 18 2025
%t A381720 Select[Range[0, 1200], IntegerQ[Surd[DigitCount[#, 2, 1], 3]] &] (* _Amiram Eldar_, Mar 05 2025 *)
%o A381720 (PARI) isok(k) = ispower(hammingweight(k), 3); \\ _Michel Marcus_, Mar 05 2025
%o A381720 (Python)
%o A381720 from itertools import count, islice, combinations
%o A381720 from sympy import integer_nthroot
%o A381720 def A381720_gen(): # generator of terms
%o A381720     a = []
%o A381720     yield 0
%o A381720     for l in count(1):
%o A381720         b = 1<<l-1
%o A381720         yield from sorted(sum(p)+b for i in range(1,integer_nthroot(l,3)[0]+1) for p in combinations(a,i**3-1))
%o A381720         a.append(b)
%o A381720 A381720_list = list(islice(A381720_gen(),53)) # _Chai Wah Wu_, Mar 06 2025
%Y A381720 Cf. A000079, A000120, A000578, A023690.
%Y A381720 Cf. A084561, A344602.
%K A381720 nonn,base
%O A381720 1,3
%A A381720 _Ctibor O. Zizka_, Mar 05 2025