A338611 Numbers all of whose divisors, excluding the divisor 1, are evil numbers (A001969) with a record number of divisors.

1, 3, 9, 15, 45, 135, 765, 2295, 196605, 589815, 12884901885, 38654705655

Offset

1,2

Comments

A number m is in this sequence if it is in A093688, and d(m) > d(k) for all terms k < m in A093688, where d(m) is the number of divisors of m (A000005).

The corresponding record numbers of divisors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, ...

Apparently, all the terms except for 1 are products of powers of Fermat primes (A019434). 3 seems to be the only prime with multiplicity larger than 1 in some of the terms. There are no other terms in this sequence that are products of powers of the 5 known Fermat primes.

Links

Table of n, a(n) for n=1..12.

Example

The first 4 terms of A093688 are 1, 3, 5, 9, and their numbers of divisors are 1, 2, 2, 3. The record values 1, 2 and 3 occur at 1, 3 and 9 that are the first 3 terms of this sequence.

Mathematica

evilQ[n_] := EvenQ @ DigitCount[n, 2, 1]; allDivEvilQ[n_] := AllTrue[Rest @ Divisors[n], evilQ]; divNumMax = 0; seq={}; Do[If[allDivEvilQ[n] && (divNum = DivisorSigma[0, n]) > divNumMax, divNumMax = divNum; AppendTo[seq, n]], {n, 1, 6*10^5}]; seq

Crossrefs

Subsequence of A001969 and A093688.

Similar sequence with odious numbers: A330289.

Cf. A000005, A019434.

Sequence in context: A171929 A188597 A330815 * A329420 A120403 A247967

Adjacent sequences: A338608 A338609 A338610 * A338612 A338613 A338614

Keyword

nonn,base,more

Author

Amiram Eldar, Nov 03 2020

Status

approved