A340638 Integers whose number of divisors that are Zuckerman numbers sets a new record.

1, 2, 4, 6, 12, 24, 72, 144, 360, 432, 1080, 2016, 2160, 6048, 8064, 15120, 24192, 48384, 88704, 120960, 241920, 266112, 532224, 1064448, 1862784, 2661120, 3725568, 5322240, 7451136, 10450944, 19160064, 20901888, 28740096, 38320128, 57480192, 99283968, 114960384

Offset

1,2

Comments

A Zuckerman number is a number that is divisible by the product of its digits (A007602).

The terms in this sequence are not necessarily Zuckerman numbers. For example a(7) = 72 has product of digits = 14 and 72/14 = 36/7 = 5.142...

The first seven terms are the first seven terms of A087997, then A087997(8) = 66 while a(8) = 144.

Links

Table of n, a(n) for n=1..37.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Example

The 8 divisors of 24 are all Zuckerman numbers, and also, 24 is the smallest integer that has at least 8 divisors that are Zuckerman numbers, hence 24 is a term.

Mathematica

zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; s[n_] := DivisorSum[n, 1 &, zuckQ[#] &]; smax = 0; seq = {}; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 10^5}]; seq (* Amiram Eldar, Jan 14 2021 *)

Prog

(PARI) isokz(n) = iferr(!(n % vecprod(digits(n))), E, 0); \\ A007602
lista(nn) = {my(m=0); for (n=1, nn, my(x = sumdiv(n, d, isokz(d)); ); if (x > m, m = x; print1(n, ", ")); ); } \\ Michel Marcus, Jan 15 2021

Crossrefs

Cf. A007602, A335037, A337941.

Subsequence of A335038.

Similar for palindromes (A093036), repdigits (A340548), repunits (A340549), Niven numbers (A340637).

Sequence in context: A340548 A087997 A355699 * A383547 A177905 A118405

Adjacent sequences: A340635 A340636 A340637 * A340639 A340640 A340641

Keyword

nonn,base

Author

Bernard Schott, Jan 14 2021

Extensions

More terms from David A. Corneth and Amiram Eldar, Jan 15 2021

Status

approved