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Positive numbers all of whose divisors are negabinary palindromes (A331891) with a record number of divisors.
0

%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