A090334 -id:A090334 - cp's OEIS frontend

A090332 Numbers with no divisors >1 that are prefixes of other divisors in binary representation.

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 47, 49, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 129, 131, 133, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179, 181

Offset: 1

Reinhard Zumkeller, Nov 26 2003

A090333 is a subsequence.
Complement of A090334.
Divisors >1 of a(n) in binary representation form a prefix code.

			Divisors >1 of a(188)=637: {7,13,49,91,637}, in binary: {111, 1101, 110001, 1011011, 1001111101}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A090330, A090331, A090333, A090335, A088862.

filter:= proc(n) local d,  i,j;
  d:= numtheory:-divisors(n);
  for i in d do
    for j from 1 to ilog2(i)-1 do
      if member(floor(i/2^j), d) then return false fi
  od od;
  true
end proc:
select(filter, [$1..200]); # Robert Israel, Jul 08 2020

filterQ[n_] := Catch@Module[{d = Divisors[n], j}, Do[
     For[j = 1, j <= Floor@Log[2, i]-1, j++,
     If[MemberQ[d, Floor[i/2^j]], Throw[False]]], {i, d}];
     True];
Select[Range[200], filterQ] (* Jean-François Alcover, Dec 15 2021, after Robert Israel *)

A090330(a(n)) = 0.

A090331(a(n)) = 1.

Missing term 121 inserted by Robert Israel, Jul 08 2020

A090330 Number of divisors >1 of n that are prefixes of other divisors of n in binary representation.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 1, 1, 3, 0, 3, 0, 3, 1, 2, 0, 5, 0, 1, 1, 3, 0, 4, 0, 4, 0, 2, 0, 6, 0, 2, 1, 5, 0, 4, 0, 3, 2, 2, 0, 7, 0, 3, 1, 3, 0, 5, 1, 5, 1, 1, 0, 7, 0, 1, 2, 5, 0, 4, 0, 4, 0, 4, 0, 8, 0, 2, 1, 4, 0, 5, 0, 7, 1, 2, 0, 7, 1, 2, 1, 5, 0, 6, 0, 3, 1, 2, 1, 9, 0, 2, 1, 5, 0, 5

Offset: 1

Reinhard Zumkeller, Nov 26 2003

nonn

a(n) = A090329(n) - 1.

Cf. A090332, A090334, A090331.

A090331 Largest proper divisor of n that is also a prefix of n in binary.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 3, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 5, 23, 1, 24, 1, 25, 3, 26, 1, 27, 1, 28, 3, 29, 1, 30, 1, 31, 7, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1, 42, 5, 43

Offset: 2

Reinhard Zumkeller, Nov 26 2003, corrected May 08 2004

a(n) = 1 iff A090330(n) = 0;

a(A090332(n))=1; a(A090334(n))>1.

Antti Karttunen, Table of n, a(n) for n = 2..65537
Eric Weisstein's World of Mathematics, Proper Divisor
Index entries for sequences related to binary expansion of n

Cf. A032742, A007088, A090330, A090332 (after its initial 1, gives the positions of 1's), A090334 (positions of terms > 1).

A090331(n) = { my(w=binary(n),x); fordiv(n,d,if(d>1, x=binary(n/d); if(w[1..#x] == x, return(n/d)))); }; \\ Antti Karttunen, Jan 24 2025

Edited by N. J. A. Sloane, Aug 19 2008 at the suggestion of Leroy Quet

cp's OEIS Frontend

A090332 Numbers with no divisors >1 that are prefixes of other divisors in binary representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A090330 Number of divisors >1 of n that are prefixes of other divisors of n in binary representation.

Views

Author

Keywords

Comments

Crossrefs

A090331 Largest proper divisor of n that is also a prefix of n in binary.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions