A090331 -id:A090331 - 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.

A090334 Numbers with at least one divisor >1 that is a prefix of another divisor in binary representation.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Reinhard Zumkeller, Nov 26 2003

Complement of A090332.

Even numbers >2 belong to this sequence.

			Divisors >1 of 51: {3,17,51}, in binary: {11, 10001, 110011}, as '11' is a prefix of '110011' 51 is a term.

Cf. A090330, A090331.

A090330(a(n)) > 0.

A090331(a(n)) > 1.

A093061 6 * Sum_{d|n} (d mod 3).

Original entry on oeis.org

6, 18, 6, 24, 18, 18, 12, 36, 6, 36, 18, 24, 12, 36, 18, 42, 18, 18, 12, 54, 12, 36, 18, 36, 24, 36, 6, 48, 18, 36, 12, 54, 18, 36, 36, 24, 12, 36, 12, 72, 18, 36, 12, 54, 18, 36, 18, 42, 18, 54, 18, 48, 18, 18, 36, 72, 12, 36, 18, 54, 12, 36, 12, 60, 36, 36, 12, 54, 18, 72, 18, 36, 12

Offset: 1

N. J. A. Sloane, Nov 02 2008

The old entry with this sequence number was a duplicate of A090331.

Cf. A004016, A082909.

Table[6 Sum[Mod[d, 3], {d, Divisors[n]}], {n, 50}] (* Wesley Ivan Hurt, May 28 2014 *)

a(n) = 6*sumdiv(n, d, d % 3); \\ Michel Marcus, May 26 2014

a(n) = 6*A082909(n). - Michel Marcus, May 26 2014

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

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

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A090334 Numbers with at least one divisor >1 that is a prefix of another divisor in binary representation.

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A093061 6 * Sum_{d|n} (d mod 3).

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