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.

Showing 1-6 of 6 results.

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

Views

Author

Reinhard Zumkeller, Nov 26 2003

Keywords

Comments

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

Crossrefs

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

Views

Author

Reinhard Zumkeller, Nov 26 2003, corrected May 08 2004

Keywords

Comments

a(n) = 1 iff A090330(n) = 0;
a(A090332(n))=1; a(A090334(n))>1.

Links

Crossrefs

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

Programs

  • PARI
    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

Extensions

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

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

Views

Author

Reinhard Zumkeller, Nov 26 2003

Keywords

Comments

Complement of A090332.
Even numbers >2 belong to this sequence.

Examples

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

Crossrefs

Cf. A090330, A090331.

Formula

A090330(a(n)) > 0.
A090331(a(n)) > 1.

A088862 Smallest number with exactly n prime factors (possibly with repetitions) such that in binary representation the divisors >1 form a prefix code.

Original entry on oeis.org

1, 2, 9, 125, 625, 3125, 15625, 78125, 9921875, 1260078125, 25937424601, 285311670611, 145223640340999, 21557022777501157, 799006685782884121, 15181127029874798299, 288441413567621167681, 5480386857784802185939
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 26 2003

Keywords

Comments

a(n) = Min{k: A090330(k)=0 and A001222(k)=n};
Conjecture: the sequence is defined for all n.
If a(n) = p^n, then for m > n, a(m) >= p^m. In particular, a(n) = 19^n for 13 < n < 86. - David Wasserman, Aug 24 2005

Examples

			a(5) = 3125 = 5^5 with divisors >1: 5, 25, 125, 625, 3125, which are prefix-free in binary: 101, 11001, 1111101, 1001110001, 110000110101.

Crossrefs

Cf. A090335, A090332.

Extensions

More terms from David Wasserman, Aug 24 2005

A090333 Nonprimes with no divisors >1 that are prefixes of other divisors in binary representation.

Original entry on oeis.org

1, 9, 25, 33, 35, 49, 65, 69, 77, 91, 121, 125, 129, 133, 141, 143, 145, 155, 161, 169, 209, 247, 259, 265, 267, 287, 289, 295, 299, 301, 305, 319, 323, 329, 341, 343, 361, 371, 377, 391, 395, 403, 407, 437, 451, 473, 481, 485, 493, 505, 515, 519, 527, 529, 533
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 26 2003

Keywords

Comments

A090332 without primes.
Includes all members of A001248 except 4. - Robert Israel, Jul 08 2020

Links

Crossrefs

Cf. A001248, A090332.

Programs

  • Maple
    filter:= proc(n) local d,  i,j;
  if isprime(n) then return false fi;
  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, [seq(i,i=1..1000,2)]); # Robert Israel, Jul 08 2020
  • Mathematica
    filterQ[n_] := If[PrimeQ[n], False, 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[1000], filterQ] (* Jean-François Alcover, Dec 16 2021, after Robert Israel *)

Extensions

Missing term 121 inserted by Robert Israel, Jul 08 2020

A090335 Smallest number with exactly n distinct prime factors such that in binary representation the divisors >1 form a prefix code.

Original entry on oeis.org

1, 2, 33, 2093, 239723, 29918977, 17597346187, 17155385292853, 10633323942990239, 15790705299133143889
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 26 2003

Keywords

Comments

Conjecture: the sequence is defined for all n.
a(10) <= 281594003343201014885689. - David Wasserman, Dec 13 2005

Examples

			a(3) = 2093 = 7*13*23 with divisors >1: 7, 13, 23, 91, 161, 299 and 2093, which are prefix-free in binary: 111, 1101, 10111, 1011011, 10100001, 100101011 and 100000101101.

Crossrefs

Cf. A088862, A090332.

Formula

a(n) = Min{k: A090330(k)=0 and A001221(k)=n}.

Extensions

More terms from David Wasserman, Dec 13 2005
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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