A090330 -id:A090330 - 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

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

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.

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

Reinhard Zumkeller, Nov 26 2003

nonn

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

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

Cf. A090335, A090332.

More terms from David Wasserman, Aug 24 2005

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

Reinhard Zumkeller, Nov 26 2003

Conjecture: the sequence is defined for all n.

a(10) <= 281594003343201014885689. - David Wasserman, Dec 13 2005

			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.

Cf. A088862, A090332.

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

More terms from David Wasserman, Dec 13 2005

A090329 Number of divisors of n that are prefixes of other divisors of n in binary representation.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 3, 1, 6, 1, 2, 2, 4, 1, 5, 1, 5, 1, 3, 1, 7, 1, 3, 2, 6, 1, 5, 1, 4, 3, 3, 1, 8, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 8, 1, 2, 3, 6, 1, 5, 1, 5, 1, 5, 1, 9, 1, 3, 2, 5, 1, 6, 1, 8, 2, 3, 1, 8, 2, 3, 2, 6, 1, 7, 1, 4, 2, 3, 2, 10, 1, 3, 2, 6, 1, 6

Offset: 1

Reinhard Zumkeller, Nov 26 2003

			Divisors of n = 35: {1,5,7,35}, in binary {1,101,111,100011}: as only '1' is a prefix, a(35) = 1;
Divisors of n = 45: {1,3,5,9,15,45}, in binary {1,11,101,1001,1111,101101}: '1' is a prefix of all other divisors, '11' of '1111' and '101' of '101101', therefore a(45) = 3.

Cf. A000005, A090330.

a(p) = 1 for all primes p.

a(n) = A090330(n) + 1.

cp's OEIS Frontend

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

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

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

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

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A090329 Number of divisors of n that are prefixes of other divisors of n in binary representation.

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