cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Reinhard Zumkeller, Jan 03 2008

Keywords

Comments

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

Links

Crossrefs

Variant of A086063.

Programs

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

A076141 Number of times n occurs as a binary sub-pattern of n^2.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 31 2002

Keywords

Comments

a(A018826(n))>0; is a(n)<=1 for all n?
Not multiplicative: a(5) = 0, a(29) = 0, a(145) = 1. - David W. Wilson, Jun 10 2005
a(n) <= 1 for n <= 10^6. - Robert Israel, Jul 11 2018

Examples

			a(27) = 1 as 27 = '11011' occurs in 27^2=729 = '1011011001' once: '**11011***'.

Links

Crossrefs

Cf. A018826.

Programs

  • Maple
    f:= proc(n) local S,S2;
    S:= convert(convert(n,binary),string);
    S2:= convert(convert(n^2,binary),string);
    nops([StringTools:-SearchAll(S,S2)])
end proc:
map(f, [$0..200]); # Robert Israel, Jul 11 2018
  • PARI
    issub(b, bs, k) = {for (i=1, #b, if (b[i] != bs[i+k-1], return (0));); return (1);}
a(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));} \\ Michel Marcus, Mar 15 2015

A162721 A positive integer k is included if, when k is represented in binary, it contains the binary representation of every distinct prime dividing k as substrings, without overlapping of the substrings.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 27, 29, 31, 32, 37, 41, 43, 47, 53, 54, 59, 61, 63, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 108, 109, 113, 125, 126, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 175, 179, 181, 191, 193, 197, 199, 211, 216, 223, 227, 229, 233, 239, 241, 243, 245, 251, 252, 256
Offset: 1

Views

Author

Leroy Quet, Jul 11 2009

Keywords

Comments

Includes all primes and powers of 2, but no even semiprimes except 4. Contains p^2 for primes p in A018826. - Robert Israel, Jan 10 2023
Contains no squarefree numbers except primes. - Robert Israel, Jan 12 2023

Examples

			20 in binary is 10100. The distinct primes dividing 20 are 2 and 5, which are 10 and 101 in binary. Both 10 and 101 occur in 10100, but with overlapping. So 20 is not in this sequence. However, 54 in binary is 110110. 54 is divisible by 2 and 3, which are 10 and 11 in binary. Both 10 and 11 occur in 110110 without overlapping. (1{10}{11}0.) So 54 is in this sequence.

Links

Crossrefs

Cf. A018826, A162722.

Programs

  • Maple
    # Requires Maple 2018 or later
satfilter:= proc(n) local n2, P, nP, X, P2, J, Cons, Clause, i,j,k, Ck;
  n2:= convert(n,base,2);
  P:= numtheory:-factorset(n);
  nP:= nops(P);
  P2:= map(convert,P,base,2);
  J:= map(t -> map(s -> [$s..s+nops(t)-1],select(i -> n2[i..i+nops(t)-1] = t, [$1..nops(n2)+1-nops(t)])), P2);
  if member([],J) then return false fi;
  Cons:= true;
  for i from 1 to nops(J) do
    Clause:= X[i,J[i][1]];
    for j from 2 to nops(J[i]) do
      Clause:= Clause &or X[i,J[i][j]]
    od;
    Cons:= Cons &and Clause;
  od;
  for k from 1 to nops(n2) do
    Ck:= {};
    for i from 1 to nP do
      for j from 1 to nops(J[i]) do if member(k,J[i,j]) then Ck:= Ck union {X[i,J[i,j]]} fi od od;
    if nops(Ck) >= 2 then for i from 2 to nops(Ck) do for j from 1 to i-1 do Cons:= Cons &and (¬(Ck[i]) &or ¬(Ck[j])) od od fi;
  od:
  Logic:-Satisfiable(Cons);
end proc:
select(satfilter, [$2..1000]); # Robert Israel, Jan 10 2023

Extensions

More terms from Sean A. Irvine, Dec 09 2010

A133408 Numbers k such that k is a substring of both its square and its cube in base 2 (written in base 10).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 41, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296
Offset: 1

Views

Author

Jonathan Vos Post, Dec 22 2007

Keywords

Comments

Binary analog of A029943. Subset of A018826.
Row 2 of array whose row 1 is A002275 and whose row 10 is A029943.
Contains every power of 2. Is 41 the only term which is not a power of 2? - Sean A. Irvine, Oct 11 2009
Up to 1.7*10^13 the sequence does not contain numbers greater than 41 which are not a power of 2. - Giovanni Resta, Aug 30 2018

Examples

			41 is a term because 41 (base 2) = 101001, which is a substring of 41^2 (base 2) = 11010010001 and which is a substring of 41^3 (base 2) = 10000110100111001.

Crossrefs

Cf. A007088, A018826, A029943.

Programs

  • Mathematica
    Select[Range[0,10^6],SequenceCount[IntegerDigits[#^2,2],IntegerDigits[#,2]]>0&&SequenceCount[IntegerDigits[#^3,2],IntegerDigits[#,2]]>0&] (* James C. McMahon, Mar 17 2025 *)

Formula

{k such that A007088(k) is a substring of A007088(k^2) and is a substring of A007088(k^3)}.

Extensions

Corrected and extended by Sean A. Irvine, Oct 11 2009
a(32) from Oliver Allen, Aug 08 2017
a(33)-a(35) from Oliver Allen, Aug 10 2017
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