%I #11 Jan 31 2020 02:19:29
%S 1,3,21,5397,353703189
%N Positive numbers all of whose divisors are negabinary palindromes (A331891) with a record number of divisors.
%C A number m is in this sequence if it is in A331896, and d(m) > d(k) for all terms k < m in A331896, where d(m) is the number of divisors of m (A000005).
%C The corresponding number of divisors are 1, 2, 4, 8, 16, ...
%C Apparently the terms are squarefree products of Mersenne primes (A000668) and Fermat primes (A019434).
%C a(6) <= 3301173437325733061894777515.
%e 21 is a term since all the divisors of 21, {1, 3, 7, 21}, are palindromes in negabinary representation: {1, 111, 11011, 10101}, and it has 4 divisors, more than the number of divisors of smaller numbers with this property: 1, 3, 5, 7, 11, and 17 have no more than 2 divisors.
%t negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]];
%t negaBinPalQ[n_] := PalindromeQ[negabin[n]];
%t negaBinAllDivPalQ[n_] := negaBinPalQ[n] && AllTrue[Most @ Divisors[n], negaBinPalQ];
%t divNumMax = 0; seq={}; Do[If[negaBinAllDivPalQ[n] && (divNum = DivisorSigma[0, n]) > divNumMax, divNumMax = divNum; AppendTo[seq, n]], {n, 1, 6000}]; seq
%Y Cf. A000005, A329420, A331891, A331896.
%K nonn,base,more
%O 1,2
%A _Amiram Eldar_, Jan 30 2020