A018826 Numbers n such that n is a substring of its square when both are written in base 2.

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

Offset

1,3

Comments

Complement of A136492. - Reinhard Zumkeller, Jan 01 2008

A136510(a(n)) = 2 for n>0. - Reinhard Zumkeller, Jan 03 2008

From Robert Israel, Jul 11 2018: (Start)

Contains A000079.

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)

Links

Giovanni Resta, Table of n, a(n) for n = 1..700 (first 200 terms from Robert Israel)

Example

27 in binary is 11011 and 27^2 = 729 in binary is 1011011001 which has substring 11011. - Michael Somos, Mar 16 2015

Maple

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:
select(filter, [$0..10000]); # Robert Israel, Jul 11 2018

Mathematica

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 *)

Prog

(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:]
print([k for k in range(9999) if ok(k)]) # Michael S. Branicky, Apr 04 2024

Crossrefs

Cf. A000079, A076141, A136490, A136492, A136510.

Cf. A018827 (base 3), A018828 (base 4), A018829 (base 5), A018830 (base 6), A018831 (base 7), A018832 (base 8), A018833 (base 9), A018834 (base 10).

Sequence in context: A180249 A060957 A322326 * A342130 A258624 A236292

Adjacent sequences: A018823 A018824 A018825 * A018827 A018828 A018829

Keyword

nonn,base

Author

David W. Wilson

Status

approved