A136490 -id:A136490 - cp's OEIS frontend

A136491 Complement of A136490.

11, 13, 19, 21, 22, 25, 26, 27, 35, 37, 38, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 59, 61, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110

Offset: 1

Reinhard Zumkeller, Jan 01 2008

Cf. A136490, A136492.

Select[Range[0, 150], StringFreeQ[IntegerString[#^3, 2], IntegerString[#, 2]] &] (* Paolo Xausa, Apr 05 2024 *)

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

Original entry on oeis.org

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

David W. Wilson

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)

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

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

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).

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

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 *)

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

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

A029942 Numbers k such that the decimal expansion of k^3 contains k as a substring.

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 10, 24, 25, 32, 40, 49, 50, 51, 56, 60, 75, 76, 90, 99, 100, 125, 240, 249, 250, 251, 375, 376, 400, 490, 499, 500, 501, 510, 600, 624, 625, 749, 750, 751, 760, 782, 875, 900, 990, 999, 1000, 1249, 1250, 2400, 2490, 2500, 2510

Offset: 1

Patrick De Geest

			24 is a term as 24^3 = 13824 contains 24 as a substring.
250 is a term as 250^3 = 1562500 contains 250 as a substring.
6^3 = 21_6, 782^3 = 4_782_11768.

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 500 terms from Reinhard Zumkeller)

Cf. A018834 (squares), A075904 (4th powers), A075905 (5th powers), A136490 (base 2).

Cf. A000578. Supersequence of A029943.

import Data.List (isInfixOf)
a029942 n = a029942_list !! (n-1)
a029942_list = [x | x <- [0..], show x `isInfixOf` show (x^3)]
-- Reinhard Zumkeller, Feb 29 2012

n3ssQ[n_]:=Module[{idn=IntegerDigits[n],idn3=Partition[ IntegerDigits[ n^3], IntegerLength[n],1]},MemberQ[idn3,idn]]; Join[{0},Select[Range[ 2600],n3ssQ]] (* Harvey P. Dale, Jan 23 2012 *)
Select[Range[0,2600],SequenceCount[IntegerDigits[#^3],IntegerDigits[ #]]> 0&] (* Harvey P. Dale, Aug 29 2021 *)

A136510 Smallest k>1 such that in binary representation n is contained in n^k.

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 5, 3, 4, 3, 3, 2, 3, 3, 6, 3, 6, 5, 3, 3, 5, 5, 2, 3, 3, 3, 3, 2, 3, 3, 6, 3, 8, 6, 3, 3, 2, 9, 4, 5, 6, 5, 5, 3, 5, 5, 4, 5, 6, 2, 5, 3, 5, 3, 6, 3, 6, 3, 3, 2, 3, 3, 6, 3, 7, 6, 10, 3, 9, 11, 5, 7, 8, 4, 5, 3, 9, 2, 8, 9, 7, 4, 6, 5, 6, 6, 3, 5, 5, 5, 5, 3, 5, 5, 3, 5, 9, 11, 7, 5

Offset: 1

Reinhard Zumkeller, Jan 03 2008

A136511(n) = n^a(n);
a(A018826(n)) = 2; 1 < a(A136490(n)) <= 3;
conjecture: a(n) is defined for all n.

Variant of A086063.

Table[Module[{k=2},While[SequenceCount[IntegerDigits[n^k,2],IntegerDigits[ n,2]]==0,k++];k],{n,110}] (* Harvey P. Dale, Aug 20 2020 *)

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A136491 Complement of A136490.

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

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A029942 Numbers k such that the decimal expansion of k^3 contains k as a substring.

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A136510 Smallest k>1 such that in binary representation n is contained in n^k.

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