A255401 Numbers n with the property that its k-th smallest divisor, for all 1 <= k <= tau(n), contains exactly k "1" digits in its binary representation.

1, 3, 5, 17, 25, 39, 57, 201, 257, 289, 291, 323, 393, 579, 1083, 2307, 7955, 8815, 9399, 12297, 12909, 13737, 36867, 40521, 43797, 50349, 65537, 66049, 66291, 66531, 68457, 80457, 98313, 160329, 196617, 197633, 230691, 299559, 599079, 786441, 922179, 1278537

Offset: 1

Author

Jaroslav Krizek, Feb 22 2015

Keywords

• nonn
• base

Comments

For n>1; a(n) is a multiple of a Fermat prime (A019434). Subsequence of A071593.

For all divisors d_k of a(n) we have A000120(d_k) = k.

Subsequence of known numbers with k divisors:

for k = 2: 3, 5, 17, 257, 65537, ... - Fermat primes (A019434);

for k = 3: 25, 289, 66049, 4295098369, ... - some square of Fermat prime;

for k = 4: 39, 57, 201, 291, 323, 393, 579, 2307, 12297, 36867, 98313, 196617, 197633, 786441, 2359299, 805306377, 3221225481, 4295229443, 9663676419, 618475290627, 19791209299971, ... - some products of two distinct primes p*q, where p is a Fermat prime (A019434) and q is a term of sequence A081091, see (Magma) - Set(Sort([n*m: n in [A019434(n)], m in [A081091(m)] | n lt m and &+Intseq(n, 2) eq 2 and &+Intseq(m, 2) eq 3 and &+Intseq(n*m, 2) eq 4]));

for k = 6: 1083 - the only number with this property < 10^7;

for k = 8: 7955, 8815, 9399, 12909, 13737, 40521, 43797, 50349, 66291, 66531, 68457, 80457, 160329, 230691, 299559, 599079, 922179, 1278537, 2396199, 2556489, ...; see (Magma) - Set(Sort([n: n in [1..1000000] | [&+Intseq(d, 2): d in Divisors(n)] eq [1,2,3,4,5,6,7,8]])).

Conjectures: 1) Sequence is infinite. 2) 8 is the maximal value of k for numbers with this property.

Numbers 805306377, 3221225481, 4295098369, 4295229443, 9663676419, 618475290627 and 19791209299971 are also terms of this sequence.

Sequence of the smallest numbers n with k divisors having these properties for k >= 1 or 0 if no solution exists or has been found: 1, 3, 25, 39, 0, 1083, 0, 7955, ...; a(5) = a(7) = 0 if there are only 5 Fermat primes. Conjecture: a(k) = 0 for k > 8.

Examples

			The divisors of 1083, expressed in base 2 and listed in ascending order as 1, 11, 10011, 111001, 101101001, 10000111011, contain 1, 2, 3, 4, 5 and 6 "1" digits, respectively.
		

Links

• Jaroslav Krizek and Amiram Eldar, Table of n, a(n) for n = 1..50 (terms 1..45 from Jaroslav Krizek)

Crossrefs

Cf. A019434, A071593, A081091.

Programs

Magma

Set(Sort([n: n in [1..1000000] | [&+Intseq(d, 2): d in Divisors(n)] eq [1..NumberOfDivisors(n)]]))

Mathematica

Select[Range[10^6], Total @ IntegerDigits[#, 2] & /@ (d = Divisors[#]) == Range @ Length[d] &] (* Amiram Eldar, Dec 29 2019 *)

PARI

isok(n) = {my(d = divisors(n)); for (i=1, #d, if (hammingweight(d[i]) != i, return (0));); return (1);} \\ Michel Marcus, Feb 22 2015