A288069 -id:A288069 - cp's OEIS frontend

A007602 Numbers that are divisible by the product of their digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 36, 111, 112, 115, 128, 132, 135, 144, 175, 212, 216, 224, 312, 315, 384, 432, 612, 624, 672, 735, 816, 1111, 1112, 1113, 1115, 1116, 1131, 1176, 1184, 1197, 1212, 1296, 1311, 1332, 1344, 1416, 1575, 1715, 2112, 2144

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

These are called Zuckerman numbers to base 10. [So-named by J. J. Tattersall, after Herbert S. Zuckerman. - Charles R Greathouse IV, Jun 06 2017] - Howard Berman (howard_berman(AT)hotmail.com), Nov 09 2008
This sequence is a subsequence of A180484; the first member of A180484 that is not a member of A007602 is 1114. - D. S. McNeil, Sep 09 2010
Complement of A188643; A188642(a(n)) = 1; A038186 is a subsequence; A168046(a(n)) = 1: subsequence of A052382. - Reinhard Zumkeller, Apr 07 2011
The terms of n digits in the sequence, for n from 1 to 14, are 9, 5, 20, 40, 117, 285, 747, 1951, 5229, 13493, 35009, 91792, 239791, 628412, 1643144, 4314987. Empirically, the counts seem to grow as 0.858*2.62326^n. - Giovanni Resta, Jun 25 2017
De Koninck and Luca showed that the number of Zuckerman numbers below x is at least x^0.122 but at most x^0.863. - Tomohiro Yamada, Nov 17 2017
The quotients obtained when Zuckerman numbers are divided by the product of their digits are in A288069. - Bernard Schott, Mar 28 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters (2005), 2nd Edition, p. 86 (see problems 44-45).

Reinhard Zumkeller and Zak Seidov, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below)
Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.
Qizheng He and Carlo Sanna, Counting numbers that are divisible by the product of their digits, arXiv:2403.14812 [math.NT], 2024.
Giovanni Resta, Zuckerman numbers, Numbers Aplenty.
Index entries for Colombian or self numbers and related sequences

Cf. A034709, A001103, A005349, A288069.
Cf. A286590 (for factorial-base analog).
Subsequence of A002796, A034838, and A055471.

import Data.List (elemIndices)
a007602 n = a007602_list !! (n-1)
a007602_list = map succ $ elemIndices 1 $ map a188642 [1..]
-- Reinhard Zumkeller, Apr 07 2011

[ n: n in [1..2144] | not IsZero(&*Intseq(n)) and IsZero(n mod &*Intseq(n)) ];  // Bruno Berselli, May 28 2011

filter:= proc(n)
local p;
p:= convert(convert(n,base,10),`*`);
p <> 0 and n mod p = 0
end proc;
select(filter, [$1..10000]); # Robert Israel, Aug 24 2014

zuckerQ[n_] := Module[{d = IntegerDigits[n], prod}, prod = Times @@ d; prod > 0 && Mod[n, prod] == 0]; Select[Range[5000], zuckerQ] (* Alonso del Arte, Aug 04 2004 *)

for(n=1,10^5,d=digits(n);p=prod(i=1,#d,d[i]);if(p&&n%p==0,print1(n,", "))) \\ Derek Orr, Aug 25 2014

from operator import mul
from functools import reduce
A007602 = [n for n in range(1,10**5) if not (str(n).count('0') or n % reduce(mul, (int(d) for d in str(n))))] # Chai Wah Wu, Aug 25 2014

A342593 Numbers m that are not the quotient of a Zuckerman number divided by the product of its digits.

Original entry on oeis.org

10, 15, 16, 20, 24, 25, 26, 30, 32, 35, 38, 39, 40, 42, 43, 47, 50, 54, 55, 58, 60, 62, 65, 70, 71, 73, 75, 78, 80, 85, 87, 90, 92, 95, 99, 100, 105, 107, 108, 110, 115, 116, 117, 119, 120, 123, 125, 127, 130, 131, 135, 137, 138, 139, 140, 141, 142, 145, 146, 147, 150, 155

Offset: 1

Bernard Schott, Mar 16 2021

The Zuckerman numbers (A007602) are the numbers that are divisible by the product of their digits (see link).
m is a term iff A056770(m) = 0.
All the multiples of 10 are terms.
Many numbers that end with 5 are terms, first exceptions < 1000: 5, 45, 255, 315, 505, ...

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Cf. A007602, A056770, A288069.

Cf. A003635 (similar for Niven numbers).

A343681 Zuckerman numbers which when divided by product of their digits, give a quotient which is also a Zuckerman number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 36, 111, 128, 135, 144, 175, 384, 672, 735, 1111, 1296, 1575, 11111, 22176, 42624, 82944, 111111, 139968, 688128, 719712, 1111111, 1161216, 1492992, 2241792, 2794176, 4136832, 4741632, 6838272, 11111111, 12171264, 13395375, 13436928

Offset: 1

Bernard Schott, Apr 26 2021

Alternative Name: Zuckerman numbers k such that k/(product of digits of k) is also a Zuckerman number. - Wesley Ivan Hurt, Apr 26 2021

All positive repunits are terms (A002275).

			24 is a Zuckerman number as 24/(2*4) = 3, 3/3 = 1 so 3 is also a Zuckerman number, and 24 is a term.
1296 is a Zuckerman number as 1296/(1*2*9*6) = 12, 12/(1*2) = 4 so 12 is also a Zuckerman number and 1296 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..160 (terms < 3*10^14)
Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Cf. A002275, A007602, A288069, A343680, A343682.

Cf. A235507 (similar, with Niven numbers).

zuckQ[n_] := (prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod]; Select[Range[10^6], zuckQ[#] && zuckQ[#/Times @@ IntegerDigits[#]] &] (* Amiram Eldar, Apr 26 2021 *)

isz(n) = my(p=vecprod(digits(n))); p && !(n % p); \\ A007602
isok(n) = isz(n) && isz(n/vecprod(digits(n))); \\ Michel Marcus, Apr 26 2021

More terms from David A. Corneth, Apr 26 2021

A359961 Smallest Zuckerman number (A007602) with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 6, 132, 3276, 27132, 1117116, 111914712, 6111417312, 1113117121116, 1112712811322112, 11171121131111172

Offset: 0

Bernard Schott, Jan 21 2023

			3276 = 2^2*3^2*7*13 is the smallest integer with 4 distinct prime factors that is also Zuckerman number as 3276 / (3*2*7*6) = 13, so a(4) = 3276.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A007602, A288069.

Similar: A060319 (Fibonacci), A083002 (oblong), A359960 (Niven).

a(n) = my(k=1); while (!(p=vecprod(digits(k))) || (k % p) || (omega(k) != n), k++); k; \\ Michel Marcus, Jan 21 2023

a(6)-a(7) from Michel Marcus, Jan 21 2023
a(8)-a(9) from Daniel Suteu, Jan 21 2023
a(10)-a(11) from Bert Dobbelaere, Jan 29 2023

A339757 a(n) is the number of Zuckerman numbers k for which k/A007954(k) = n, where A007954(k) is the product of the decimal digits of k.

Original entry on oeis.org

9, 1, 2, 1, 1, 1, 1, 2, 3, 0, 1, 2, 3, 1, 0, 0, 1, 3, 2, 0, 1, 2, 3, 0, 0, 0, 1, 2, 2, 0

Offset: 1

Michel Marcus, Apr 04 2021

The indices of 0's are A342593.

			The integers k=1 to 9 are the Zuckerman numbers that satisfy k/A007954(k)=1, so a(1)=9.

Cf. A007954 (product of decimal digits), A007602 (Zuckerman numbers), A056770.
Cf. A288069 (Zuckerman quotients), A342593 (Zuckerman non-quotients).
Cf. A343036, A343050.

A342941 Numbers not ending with 0, that are not the quotient of a Zuckerman number divided by the product of its digits.

Original entry on oeis.org

15, 16, 24, 25, 26, 32, 35, 38, 39, 42, 43, 47, 54, 55, 58, 62, 65, 71, 73, 75, 78, 85, 87, 92, 95, 99, 105, 107, 108, 115, 116, 117, 119, 123, 125, 127, 131, 135, 137, 138, 139, 141, 142, 145, 146, 147, 155, 165, 175, 176, 178, 179, 181, 185, 189, 191, 193, 195, 197, 199

Offset: 1

Bernard Schott, Mar 30 2021

Zuckerman numbers (A007602) are the numbers that are divisible by the product of their digits (see link).

Multiples of 10 are never the quotient of a Zuckerman number divided by the product of its digits, but they are not present in this sequence (see A342593).

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Equals A342593 \ A008592.
Subsequence of A067251.
Cf. A007602, A056770, A288069.

A343050 Zuckerman numbers (A007602) ordered by increasing value of k/A007954(k) where A007954(k) is the product of the decimal digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 15, 24, 384, 175, 12, 735, 128, 672, 135, 144, 1575, 11, 1296, 139968, 624, 3276, 1886976, 224, 816, 216, 432, 34992, 1197, 12768, 315, 132, 3168, 115, 6624, 8832, 2916, 1176, 1344, 3915, 739935

Offset: 1

Michel Marcus, Apr 03 2021

a(n) is the Zuckerman number corresponding to A343036(n).

			As a table, sequence begins:
   1 [1, 2, 3, 4, 5, 6, 7, 8, 9]
   2 [36]
   3 [15, 24]
   4 [384]
   5 [175]
   6 [12]
   7 [735]
   8 [128, 672]
   9 [135, 144, 1575]
  10 []
  11 [11]
  12 [1296, 139968]
  13 [624, 3276, 1886976]
  14 [224]
  15 []
  16 []
  17 [816]
  18 [216, 432, 34992]
  19 [1197, 12768]
  20 []
  21 [315]
  22 [132, 3168]
  23 [115, 6624, 8832]
  24 []
  25 []
  26 []
  27 [2916]
  28 [1176, 1344]
  29 [3915, 739935]
  30 []
  ... where the 1st column is A056770 and the number of terms per rows is A339757.

Cf. A007954 (product of decimal digits), A007602 (Zuckerman numbers), A056770.
Cf. A288069 (Zuckerman quotients), A342593 (Zuckerman non-quotients), A343036.
Cf. A339757.

a(29)-a(45) from David A. Corneth, Apr 03 2021

A343682 Zuckerman numbers which when divided by the product of their digits, give a quotient which is a Niven (Harshad) number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 24, 36, 111, 128, 135, 144, 175, 216, 315, 384, 432, 672, 735, 1296, 1575, 2916, 11115, 11232, 11664, 12132, 12288, 12312, 13212, 13824, 14112, 16416, 22176, 23112, 23328, 26112, 27216, 31212, 32832, 34272, 34992, 42624, 72128, 77175

Offset: 1

Bernard Schott, Apr 26 2021

Repunit R(k) is a term iff k divides R(k) (A014950).

			36 is a Zuckerman number as 36/(3*6) = 2, 2/2 = 1 that is a Niven number, and 36 is a term.
315 is a Zuckerman number as 315/(3*1*5) = 21, 21/(2+1) = 7 that is a Niven number, and 315 is a term.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Cf. A005349, A007602, A235507, A288069, A343680, A343681.

nivenQ[n_] := IntegerQ[n] && (sum = Plus @@ IntegerDigits[n]) > 0 && Divisible[n, sum]; Select[Range[10^5], (prod = Times @@ IntegerDigits[#]) > 0 && nivenQ[# / prod] &] (* Amiram Eldar, Apr 26 2021 *)

isn(n) = !(n%sumdigits(n)); \\ A005349
isz(n) = my(p=vecprod(digits(n))); p && !(n % p); \\ A007602
isok(n) = isz(n) && isn(n/vecprod(digits(n))); \\ Michel Marcus, Apr 26 2021

More terms from Michel Marcus, Apr 26 2021

A360075 a(n) is the product of the digits of A007602(n), the n-th Zuckerman number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 8, 18, 1, 2, 5, 16, 6, 15, 16, 35, 4, 12, 16, 6, 15, 96, 24, 12, 48, 84, 105, 48, 1, 2, 3, 5, 6, 3, 42, 32, 63, 4, 108, 3, 18, 48, 24, 175, 35, 4, 32, 24, 108, 3, 18, 144, 21, 252, 18, 135, 8, 64, 96, 96, 288, 108, 14, 63

Offset: 1

Rémy Sigrist, Jan 24 2023

Zuckerman numbers (A007602) correspond to numbers divisible by the product of their digits.

			For n = 1515:
- A007602(1515) = 11834112,
- so a(1515) = 1*1*8*3*4*1*1*2 = 192.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A007602, A007954, A051801, A288069, A325454.

{ for (n=1, 7119, p=vecprod(digits(n)); if (p && n%p==0, print1 (p", "))) }

a(n) = A007954(A007602(n)).
a(n) = A051801(A007602(n)).
a(n) * A288069(n) = A007602(n).

A343744 Zuckerman numbers which divided by the product of their digits give integers which are also divisible by the product of their digits, and so on, until result is 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 24, 36, 128, 135, 144, 175, 384, 672, 735, 1296, 1575, 82944, 139968, 1492992, 27869184

Offset: 1

Bernard Schott, Apr 27 2021

Repunits >= 11 (A002275) are not in the sequence because, as they are fixed points of this map, they don't fit the definition.
Question: is this sequence finite as the similar sequence with Niven numbers (A114440) that has 15095 terms?
No other terms up to 2*10^9. - Michel Marcus, Apr 27 2021
From David A. Corneth, Apr 27 2021: (Start)
Terms are 7-smooth. Any prime factor > 7 will not be divided away by dividing by product of digits.
Any number k > a(26)*10^163 with product of digits vp > 0 has k/vp > a(26) so it suffices to check all candidates <= a(26)*10^163. Doing so gives no more terms so this sequence is finite and full. (End)
The number of steps needed to reach 1, has a maximum of 3, which occurs for n = 21, 23..26. - A.H.M. Smeets, Apr 29 2021

			The integer 1296 is divisible by the product of its digits as 1296/(1*2*9*6) = 12, then 12/(1*2) = 6 and 6/6 = 1; hence, 1296 is a term of this sequence.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Cf. A007602, A288069.
Cf. A114440 (similar for Harshad numbers).
Subsequence of A002473 and of A343681.

f[n_] := If[(prod = Times @@ IntegerDigits[n]) > 0 && Divisible[n, prod], n/prod, 0]; Select[Range[10^5], FixedPointList[f, #][[-1]] == 1 &] (* Amiram Eldar, Apr 27 2021 *)

isz(n) = my(p=vecprod(digits(n))); p && !(n % p); \\ A007602
isok(n) = if (n==1, return(1)); my(m=n); until(m==1, if (isz(m), my(nm = m/vecprod(digits(m))); if (nm==m, return (0), m = nm), return(0))); return(1); \\ Michel Marcus, Apr 27 2021

def proddigit(n):
    p = 1
    while n > 0:
        n, p = n//10, p*(n%10)
    return p
n, a = 1, 1
while n > 0:
    aa, pa = a, proddigit(a)
    while pa > 1 and aa%pa == 0 and aa > 1:
        aa = aa//pa
        pa = proddigit(aa)
    if aa == 1:
        print(n,a)
        n = n+1
    a = a+1 # A.H.M. Smeets, Apr 29 2021

a(26) from Michel Marcus, Apr 27 2021

cp's OEIS Frontend

A007602 Numbers that are divisible by the product of their digits.

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A342593 Numbers m that are not the quotient of a Zuckerman number divided by the product of its digits.

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A343681 Zuckerman numbers which when divided by product of their digits, give a quotient which is also a Zuckerman number.

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A359961 Smallest Zuckerman number (A007602) with exactly n distinct prime factors.

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A339757 a(n) is the number of Zuckerman numbers k for which k/A007954(k) = n, where A007954(k) is the product of the decimal digits of k.

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A342941 Numbers not ending with 0, that are not the quotient of a Zuckerman number divided by the product of its digits.

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A343050 Zuckerman numbers (A007602) ordered by increasing value of k/A007954(k) where A007954(k) is the product of the decimal digits of k.

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A343682 Zuckerman numbers which when divided by the product of their digits, give a quotient which is a Niven (Harshad) number.

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A360075 a(n) is the product of the digits of A007602(n), the n-th Zuckerman number.