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A352084 Integers m such that wt(m) divides wt(m^2) where wt(m) = A000120(m) is the binary weight of m.

%I A352084 #44 Oct 07 2022 23:59:42
%S A352084 1,2,3,4,6,7,8,12,14,15,16,21,24,28,30,31,32,37,42,45,48,53,56,60,62,
%T A352084 63,64,69,73,74,79,81,83,84,90,91,96,106,112,120,124,126,127,128,133,
%U A352084 137,138,141,146,148,155,157,158,159,161,162,165,166,168,177,180
%N A352084 Integers m such that wt(m) divides wt(m^2) where wt(m) = A000120(m) is the binary weight of m.
%C A352084 Integers m such that A000120(m) divides A159918(m).
%C A352084 This is a problem proposed by the French site Diophante in the links section.
%C A352084 The first 18 terms are the same as A268415, then A268415(19) = 41 while a(19) = 42.
%C A352084 The corresponding quotients are in A352085.
%C A352084 The smallest term k such that the corresponding quotient = n is A352086(n).
%C A352084 Some subsequences:
%C A352084 -> wt(m^2) = wt(m) iff m is in A077436.
%C A352084 -> wt(m^2) / wt(m) = 2 iff m is in A083567.
%C A352084 -> When m is a power of 2 (A000079): wt(2^k) = wt((2^k)^2) = wt(2^(2k)) = 1.
%H A352084 Martin Ehrenstein, <a href="/A352084/b352084.txt">Table of n, a(n) for n = 1..10000</a>
%H A352084 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/4786-a1730-des-chiffres-a-sommer-pour-un-entier">A1730 - Des chiffres à sommer pour un entier</a> (in French).
%e A352084 37_10 = 100101_2, digsum_2(37) = 1+1+1 = 3; then 37^2 = 1369_10 = 10101011001_2, digsum_2(1369) = 1+1+1+1+1+1 = 6; as 3 divides 6, 37 is a term.
%t A352084 Select[Range[180], Divisible[Total[IntegerDigits[#^2, 2]], Total[IntegerDigits[#, 2]]] &] (* _Amiram Eldar_, Mar 03 2022 *)
%o A352084 (Python)
%o A352084 def ok(n): return n > 0 and bin(n**2).count('1')%bin(n).count('1') == 0
%o A352084 print([m for m in range(1, 200) if ok(m)]) # _Michael S. Branicky_, Mar 03 2022
%o A352084 (PARI) isok(m) = !(hammingweight(m^2) % hammingweight(m)); \\ _Michel Marcus_, Mar 03 2022
%Y A352084 Cf. A000120, A159918, A268415, A351650, A352085, A352086.
%Y A352084 Cf. A351650 (similar for base 10).
%Y A352084 Subsequences: A000079, A023758, A077436, A083567.
%K A352084 nonn,base
%O A352084 1,2
%A A352084 _Bernard Schott_, Mar 03 2022
%E A352084 More terms from _Amiram Eldar_, Mar 03 2022