A342593 -id:A342593 - cp's OEIS frontend

A288069 Quotients obtained when the Zuckerman numbers are divided by the product of their digits.

1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 6, 3, 3, 2, 111, 56, 23, 8, 22, 9, 9, 5, 53, 18, 14, 52, 21, 4, 18, 51, 13, 8, 7, 17, 1111, 556, 371, 223, 186, 377, 28, 37, 19, 303, 12, 437, 74, 28, 59, 9, 49, 528, 67, 93, 27, 1037, 174, 22, 151, 13, 184, 29, 514, 66, 46

Offset: 1

Bernard Schott, Jun 05 2017

The Zuckerman numbers (A007602) are the numbers that are divisible by the product of their digits.
Question: Is A067251 a subsequence? No, it appears in A056770 that not all integers other than multiples of 10 can be obtained as quotient, such as 15, 16, 24, 25, 26, 32, .... (see A342941).
The limit of the sequence is infinite: for any x, there is some N such that, for all n > N, a(n) > x. Proof: a Zuckerman number with d digits is at least 10^(d-1) and has a digit product at most 9^d and so has a quotient at least 10^(d-1)/9^d which goes to infinity with d. - Charles R Greathouse IV, Jun 05 2017
The repunits A002275 are a subsequence. - Peter Schorn, Apr 05 2025

			a(11) = 12/(1*2) = 6; a(13) = 24/(2*4) = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A007602, A067251, A056770, A342593, A342941, A002275.

f:= proc(n) local L,p;
   p:= convert(convert(n,base,10),`*`);
   if p > 0 then
     if n mod p = 0 then return n/p fi
   fi
end proc:
map(f, [$1..10^4]); # Robert Israel, Jun 05 2017

Select[Table[n/Max[Times@@IntegerDigits[n],Pi/100],{n,5000}],IntegerQ] (* Harvey P. Dale, Aug 16 2021 *)

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

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A288069 Quotients obtained when the Zuckerman numbers are divided by the product of their digits.

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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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