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A018826
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Numbers n such that n is a substring of its square when both are written in base 2.
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15
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0, 1, 2, 4, 8, 16, 27, 32, 41, 54, 64, 82, 108, 128, 145, 164, 165, 256, 283, 290, 328, 487, 512, 545, 566, 580, 974, 1024, 1090, 1132, 1160, 1773, 1948, 2048, 2113, 2180, 2320, 2701, 3546, 3896, 4096, 4226, 4261, 4360, 4757, 5402, 7092, 7625, 8079, 8192
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OFFSET
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1,3
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COMMENTS
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If x satisfies x^2 == 8*x + 1 (mod 2^m) and 0 < x < 2^(m-3) then x is in the sequence. Note that x^2 == 8*x + 1 has 4 solutions mod 2^m for m >= 3. Terms obtained in this way include 27, 283, 1773, 9965, 55579, 206573, .... (End)
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LINKS
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EXAMPLE
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27 in binary is 11011 and 27^2 = 729 in binary is 1011011001 which has substring 11011. - Michael Somos, Mar 16 2015
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MAPLE
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filter:= proc(n) local S, S2;
S:= convert(convert(n, binary), string);
S2:= convert(convert(n^2, binary), string);
StringTools:-Search(S, S2)<>0
end proc:
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MATHEMATICA
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Select[Range[0, 8192], {} != SequencePosition @@ IntegerDigits[{#^2, #}, 2] &] (* Giovanni Resta, Aug 20 2018 *)
Select[Range[0, 10000], SequenceCount[IntegerDigits[#^2, 2], IntegerDigits[#, 2]]>0&] (* Harvey P. Dale, May 03 2022 *)
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PROG
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(PARI) issub(b, bs, k) = {for (i=1, #b, if (b[i] != bs[i+k-1], return (0)); ); return (1); }
a076141(n) = {if (n, b = binary(n), b = [0]); if (n, bs = binary(n^2), bs = [0]); sum(k=1, #bs - #b +1, issub(b, bs, k)); }
lista(nn) = for (n=0, nn, if (a076141(n) == 1, print1(n, ", "))); \\ Michel Marcus, Mar 15 2015
(Python)
def ok(n): return bin(n)[2:] in bin(n**2)[2:]
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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