OFFSET
0,2
COMMENTS
Note that the terms of A260986 with n > 1 can all be found here. Terms here that are not in A260986 have the property not to be a record value of the ratio (binary weight k) / (binary weight k^2).
Observation: The difference of two neighboring terms is a multiple of 2^(number of the ones after the last zero in binary expression of the smaller term).
LINKS
K. B. Stolarsky, The binary digits of a power, Proc. Amer. Math. Soc. 71 (1978), pp. 1-5.
EXAMPLE
-----------------------------------------------------------------------------
n k k^2 binary k binary k^2
-----------------------------------------------------------------------------
0 0 0 0 0
1 23 529 10111 1000010001
2 111 12321 1101111 11000000100001
3 479 229441 111011111 111000000001000001
4 1471 2163841 10110111111 1000010000010010000001
5 6015 36180225 1011101111111 10001010000001000100000001
6 24319 591413761 101111011111111 100011010000000100001000000001
7 28415 807412225 110111011111111 110000001000000010001000000001
8 114175 13035930625 11011110111111111 1100001001000000001000010000000001
MATHEMATICA
a[0] = 0; a[n_] := a[n] = Module[{step = If[n == 1, 1, 2^Length[Split[IntegerDigits[a[n - 1], 2]][[-1]]]], k = a[n - 1]}, While[DigitCount[k, 2, 1] - DigitCount[k^2, 2, 1] != n, k += step]; k]; Array[a, 23, 0] (* Amiram Eldar, Oct 14 2022 *)
PROG
(PARI) a(n) = my(k=0); while(hammingweight(k) - hammingweight(k^2) != n, k++); k; \\ Michel Marcus, Oct 14 2022
(Python)
A356877 = [0]
for n in range(1, 29):
s, k = -1, A356877[-1]
while bin(A356877[-1])[s] == "1": s -= 1
while bin(k)[2:].count("1")-bin(k**2)[2:].count("1") != n: k += 2**(abs(s)-1)
A356877.append(k)
print(A356877)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Karl-Heinz Hofmann, Oct 10 2022
STATUS
approved