cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A335038 a(n) is the smallest number m with exactly n divisors that are Zuckerman numbers, or -1 if there is no such m.

Original entry on oeis.org

1, 2, 4, 6, 18, 12, 84, 24, 168, 72, 144, 360, 432, 1080, 2016, 2160, 6048, 8064, 15120, 34272, 24192, 60480, 48384, 88704, 120960, 354816, 241920, 483840, 665280, 266112, 798336, 532224, 1596672, 1064448, 1862784, 2661120, 3725568, 5322240, 10644480, 7451136
Offset: 1

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Author

Bernard Schott, Jun 03 2020

Keywords

Comments

Inspired by A333456.
A Zuckerman number (A007602) is a number that is divisible by the product of its digits; e.g., 24 is a Zuckerman number because it is divisible by 2*4=8.
The divisors 1 and m (if m is itself a Zuckerman number) are included.
Conjecture: m always exists.
Not all terms in the sequence are Zuckerman numbers. For example a(7) = 84 has product of digits = 32 and 84/32 = 21/8 = 2.625.

Examples

			Of the six divisors of 18, five are Zuckerman numbers: 1, 2, 3, 6 and 9, and there is no smaller number with five Zuckerman divisors, hence a(5) = 18.

Links

Crossrefs

Cf. A007602, A335037, A333456 (similar, with Niven divisors).

Programs

Extensions

More terms from Amiram Eldar, Jun 03 2020
Edited, added escape clause. - N. J. A. Sloane, Jun 04 2020

A337941 Numbers whose divisors are all Zuckerman numbers (A007602).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 1111111111111111111, 11111111111111111111111
Offset: 1

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Author

Bernard Schott, Oct 01 2020

Keywords

Comments

Inspired by A337741.
Zuckerman numbers are numbers that are divisible by the product of their digits (see link).
The next term is the repunit prime R_317 which is too large to include in the data.
Primes in this sequence are 2, 3, 5, 7 and all the repunit primes (see A004023).
This sequence is infinite if and only if there are infinitely many repunit primes.

Examples

			6 is a term since all the divisors of 6, i.e., 1, 2, 3 and 6, are Zuckerman numbers.

Links

Crossrefs

Subsequence of A007602.
Similar sequences: A062687, A190217, A308851, A329419, A337741.
Cf. A004023, A335037, A335038.
Cf. A004022 (subsequence of prime repunits).

Programs

  • Mathematica
    zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; Select[Range[24], AllTrue[Divisors[#], zuckQ] &] (* Amiram Eldar, Oct 01 2020 *)
  • PARI
    isok(m) = {fordiv(m, d, my(p=vecprod(digits(d))); if (!p || (d % p), return (0))); return (1);} \\ Michel Marcus, Oct 05 2020

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

Original entry on oeis.org

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

Views

Author

Bernard Schott, Jan 14 2021

Keywords

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

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

A360073 a(n) is the greatest divisor of n divisible by the product of its own digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 5, 11, 12, 1, 7, 15, 8, 1, 9, 1, 5, 7, 11, 1, 24, 5, 2, 9, 7, 1, 15, 1, 8, 11, 2, 7, 36, 1, 2, 3, 8, 1, 7, 1, 11, 15, 2, 1, 24, 7, 5, 3, 4, 1, 9, 11, 8, 3, 2, 1, 15, 1, 2, 9, 8, 5, 11, 1, 4, 3, 7, 1, 36, 1, 2, 15, 4, 11, 6, 1, 8, 9
Offset: 1

Views

Author

Rémy Sigrist, Jan 24 2023

Keywords

Comments

Numbers divisible by the product of their digits are called Zuckerman numbers (A007602).

Examples

			For n = 10:
- the divisors of 10 are 1, 2, 5 and 10,
- 5 is divisible by 5 whereas 10 is not divisible by 1*0,
- so a(10) = 5.

Links

Crossrefs

Cf. A007602, A335037, A337941, A360074.

Programs

  • PARI
    a(n) = { fordiv (n, d, my (t=n/d, p=vecprod(digits(t))); if (p && t%p==0, return (t))) }

Formula

a(n) = n iff n belongs to A007602.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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