A331897 Positive numbers all of whose divisors are negabinary palindromes (A331891) with a record number of divisors.

1, 3, 21, 5397, 353703189

Offset

1,2

Comments

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).

The corresponding number of divisors are 1, 2, 4, 8, 16, ...

Apparently the terms are squarefree products of Mersenne primes (A000668) and Fermat primes (A019434).

a(6) <= 3301173437325733061894777515.

Links

Table of n, a(n) for n=1..5.

Example

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.

Mathematica

negabin[n_] := negabin[n] = If[n==0, 0, negabin[Quotient[n-1, -2]]*10 + Mod[n, 2]];
negaBinPalQ[n_] := PalindromeQ[negabin[n]];
negaBinAllDivPalQ[n_] := negaBinPalQ[n] && AllTrue[Most @ Divisors[n], negaBinPalQ];
divNumMax = 0; seq={}; Do[If[negaBinAllDivPalQ[n] && (divNum = DivisorSigma[0, n]) > divNumMax, divNumMax = divNum; AppendTo[seq, n]], {n, 1, 6000}]; seq

Crossrefs

Cf. A000005, A329420, A331891, A331896.

Sequence in context: A265637 A054739 A117508 * A001139 A359570 A341086

Adjacent sequences: A331894 A331895 A331896 * A331898 A331899 A331900

Keyword

nonn,base,more

Author

Amiram Eldar, Jan 30 2020

Status

approved