A255401 - OEIS

%I #28 Oct 29 2024 15:37:04

%S 1,3,5,17,25,39,57,201,257,289,291,323,393,579,1083,2307,7955,8815,

%T 9399,12297,12909,13737,36867,40521,43797,50349,65537,66049,66291,

%U 66531,68457,80457,98313,160329,196617,197633,230691,299559,599079,786441,922179,1278537

%N 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.

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

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

%C Subsequence of known numbers with k divisors:

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

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

%C 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]));

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

%C 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]])).

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

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

%C 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.

%H Jaroslav Krizek and Amiram Eldar, <a href="/A255401/b255401.txt">Table of n, a(n) for n = 1..50</a> (terms 1..45 from Jaroslav Krizek)

%e 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.

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

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

%o (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

%Y Cf. A019434, A071593, A081091.

%K nonn,base

%O 1,2

%A _Jaroslav Krizek_, Feb 22 2015