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

Author

Bernard Schott, Jan 14 2021

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

• Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Crossrefs

Cf. A007602, A335037, A337941.

Subsequence of A335038.

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

Programs

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 *)

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

Extensions

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