A038367 -id:A038367 - cp's OEIS frontend

A062045 Positive numbers whose product of digits is 12 times their sum.

666, 1479, 1497, 1568, 1586, 1658, 1685, 1749, 1794, 1856, 1865, 1947, 1974, 2349, 2394, 2439, 2446, 2464, 2493, 2644, 2934, 2943, 3249, 3294, 3345, 3354, 3429, 3435, 3453, 3492, 3534, 3543, 3924, 3942, 4179, 4197, 4239, 4246, 4264, 4293, 4329, 4335, 4353, 4392

Offset: 1

Amarnath Murthy, Jun 28 2001

			2349 belongs to the sequence as (2*3*4*9)/(2+3+4+9) = 216/18 = 12.

Harry J. Smith, Table of n, a(n) for n = 1..500

Subsequence of A061013 and hence A038367 and A052382.

Cf. A011540, A034710, A062034, A062035, A062036, A062382, A062037, A062384, A062040, A062041, A062043.

okQ[n_]:=Module[{idn=IntegerDigits[n]},Times@@idn/Total[idn]==12]
Select[Range[10000],okQ] (* Harvey P. Dale, Nov 25 2010 *)

isok(n) = my(d=digits(n)); vecprod(d)==12*vecsum(d) \\ Mohammed Yaseen, Sep 12 2022

from math import prod
def ok(n): d = list(map(int, str(n))); return prod(d) == 12*sum(d)
print([k for k in range(1, 4400) if ok(k)]) # Michael S. Branicky, Sep 12 2022

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jul 06 2001
More terms from Harvey P. Dale, Nov 25 2010
Offset corrected by Mohammed Yaseen, Sep 12 2022

A061013 Numbers k such that (product of digits of k) is divisible by (sum of digits of k), where 0's are not permitted.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 36, 44, 63, 66, 88, 123, 132, 138, 145, 154, 159, 167, 176, 183, 189, 195, 198, 213, 224, 231, 235, 242, 246, 253, 257, 264, 268, 275, 279, 286, 297, 312, 318, 321, 325, 333, 345, 347, 352, 354, 357, 369, 374, 375, 381, 396, 415

Offset: 1

Heiner Muller-Merbach (hmm(AT)sozwi.uni-kl.de), Jun 06 2001

Called "perfect years". 1998 and 2114 are the nearest past and future examples.

			1998 is perfect since 1*9*9*8/(1+9+9+8) = 24.

H. Herles, Reformstau, Gefuehlsstau, Verkehrsstau. Generalanzeiger, 12/31/1997, p. V.
H. Muller-Merbach and L. Logelix, Perfekte Jahre, Technologie und Management, Vol. 42, 1993, No. 1, p. 47 and No. 2, p. 95.

David A. Corneth, Table of n, a(n) for n = 1..10000
H. Muller-Merbach, Wunsche für das "perfekte Jahr" 1998

See A038367 for case where 0 digits are allowed. Cf. A055931.

Cf. A274124.

for n from 1 to 3000 do a := convert(n,base,10):s := add(a[i],i=1..nops(a)):p := mul(a[i],i=1..nops(a)): if p<>0 and p mod s=0 then printf(`%d,`,n):fi:od:

Select[Range[415], FreeQ[x = IntegerDigits[#], 0] && Divisible[Times @@ x, Plus @@ x] &] (* Jayanta Basu, Jul 13 2013 *)

is(n) = my(d = digits(n)); vd = vecprod(d); vd != 0 && vd % vecsum(d) == 0 \\ David A. Corneth, Mar 15 2021

from math import prod
def ok(n):
    d = list(map(int, str(n)))
    pod, sod = prod(d), sum(d)
    return pod and pod%sod == 0
print([k for k in range(416) if ok(k)]) # Michael S. Branicky, Mar 28 2022

More terms from Larry Reeves (larryr(AT)acm.org) and Vladeta Jovovic, Jun 07 2001

A274124 Numbers n such that (product of digits of n) is divisible by (sum of digits of n) and digits of n are in nondecreasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 36, 44, 66, 88, 123, 138, 145, 159, 167, 189, 224, 235, 246, 257, 268, 279, 333, 345, 347, 357, 369, 448, 456, 459, 466, 578, 579, 666, 678, 789, 999, 1124, 1146, 1168, 1225, 1233, 1236, 1247, 1258, 1269, 1344, 1348, 1356, 1368, 1447

Offset: 1

David A. Corneth, Jun 10 2016

Every number with a digit 0 is in A038367. Every permutation of every element of this is in A038367. These elements describe A038367 completely.

Intersection of A038367 and A009994.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A038367, A009994.

is(n) = my(v=vecsort(digits(n))); v==digits(n) && prod(i=1,#v,v[i]) % vecsum(v)==0

A055931 Product of the digits of n divides the sum of the digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 111, 112, 121, 123, 132, 211, 213, 231, 312, 321, 1111, 1113, 1124, 1131, 1142, 1214, 1241, 1311, 1412, 1421, 2114, 2141, 2411, 3111, 4112, 4121, 4211, 11111, 11112, 11114, 11121, 11125, 11133, 11141, 11152, 11211, 11215

Offset: 1

Robert G. Wilson v, Jul 17 2000

Cf. A038367.

Do[If[Mod[Apply[Plus, IntegerDigits[n]], Apply[Times, IntegerDigits[n]]]==0, Print[n]], {n, 1, 100000}]
pdsQ[n_]:=Module[{idn=IntegerDigits[n]},!MemberQ[idn,0] && Divisible[ Total[idn],Times@@idn]]; Select[Range[12000],pdsQ] (* Harvey P. Dale, Dec 07 2011 *)

A138566 Numbers n whose digits satisfy the following Diophantine equation: P = (X1* ... *Xr)/(X1+ ... +Xr), where P = prime number, Xi = digits of n.

Original entry on oeis.org

36, 44, 63, 66, 138, 145, 154, 159, 167, 176, 183, 195, 224, 235, 242, 253, 257, 275, 279, 297, 318, 325, 333, 345, 352, 354, 357, 375, 381, 415, 422, 435, 451, 453, 514, 519, 523, 527, 532, 534, 537, 541, 543, 572, 573, 591, 617, 671, 716, 725, 729, 735

Offset: 1

Ctibor O. Zizka, May 12 2008

			n = 761, we have (7*6*1)/(7+6+1) = 3; a(56)= 761.
n = 792, we have (7*9*2)/(7+9+2) = 7; a(57)= 792.
n = 813, we have (8*1*3)/(8+1+3) = 2; a(58)= 813.

Subsequence of A038367.

Cf. A137923.

Block[{i = k, r = {}}, r = Table[p = IntegerDigits@i; If[PrimeQ[Times @@ p/Total@p], k, Nothing], {k, 735}];r] (* Mikk Heidemaa, May 26 2024 *)

A240520 Numbers that set a new integer record for the ratio between the product and the sum of their digits.

Original entry on oeis.org

1, 36, 66, 88, 257, 268, 279, 369, 459, 578, 579, 678, 789, 999, 2589, 2688, 2799, 3699, 3789, 4599, 4689, 4789, 5788, 5889, 7889, 8888, 18999, 25889, 26789, 26888, 27788, 28899, 37899, 38889, 45999, 46899, 47799, 47889, 55899, 56889, 57789, 58999, 78999, 257899

Offset: 1

Paolo P. Lava, Apr 07 2014

			For n = 2 we have 2 / 2 = 1.
For n = 36 we have 18 / 9 = 2.
For n = 66 we have 36 / 12 = 3.
For n = 88 we have 64 / 16 = 4.
Etc.

Paolo P. Lava and Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 100 terms from Paolo P. Lava)

A038367.

S:=proc(s) local j,w; w:=convert(s,base,10); sum(w[j],j=1..nops(w)); end:
U:=proc(s) local j,w; w:=convert(s,base,10); mul(w[j],j=1..nops(w)); end:
P:=proc(q) local a,n; a:=0;
for n from 2 to q do if type(U(n)/S(n),integer) then if U(n)/S(n)>a then
a:=U(n)/S(n); print(n); fi; fi; od; end: P(10^6);

A371631 Primes whose product of nonzero digits divided by the sum of its digits is also prime.

Original entry on oeis.org

167, 257, 523, 541, 617, 761, 1447, 1607, 1861, 2053, 2251, 2503, 2521, 2851, 4051, 5023, 5281, 5821, 6701, 8161, 8521, 10067, 10607, 10861, 11273, 11471, 12713, 13127, 13217, 13721, 14407, 16007, 17123, 17231, 17321, 18061, 20507, 20521, 21247, 21317, 21713, 22051

Offset: 1

Mikk Heidemaa, May 24 2024

No term N can have a "9" digit. [Proof: The sum of the digits of N is not a multiple of 3, but the numerator would be a multiple of 9, and so the number would be a multiple of 9, so not a prime.]

			167 (prime) is a term because 1*6*7/(1+6+7)=42/14=3 (prime).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A038367.

Equals prime terms of A138566.

pQ[n_] := Block[{idp = DeleteCases[IntegerDigits[n], 0]}, PrimeQ[Times @@ idp/Total@ idp]]; Cases[Prime@ Range@ PrimePi[10^5], _?pQ]
Select[Prime[Range[2500]],PrimeQ[Times@@(IntegerDigits[#]/.(0->1))/Total[ IntegerDigits[ #]]]&] (* Harvey P. Dale, Sep 24 2024 *)

A381631 Numbers k such that the product of k and its digits is divisible by the sum of its digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 22, 24, 26, 27, 30, 36, 40, 42, 44, 45, 48, 50, 54, 60, 62, 63, 66, 70, 72, 80, 81, 84, 88, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 114, 116, 117, 120, 123, 126, 130, 132, 133, 134, 135, 138, 140

Offset: 1

Jakub Buczak, Mar 02 2025

Positive integers with the digit 0 (see A011540) are terms, since the product of it and its digits is A098736(k) = 0 which is divisible by any sum of digits.
Terms with a 0 digit form various runs of consecutive terms, such as from 100...00 through to 111...10.
Terms without a 0 digit can form runs of 9 terms: see A381697.
A prime > 7 is never divisible by its sum of digits (because the sum is smaller than the prime) so that primes > 7 occur in this sequence only when their product of digits is divisible by sum of digits (the primes in A038367).

			36 is a term because 36*3*6 is divisible by 3+6.
140 is a term because 140*1*4*0 equals 0, which is trivially divisible by 1+4+0.

Cf. A007953, A098736, A381697.

Cf. A011540, A038367, A005349.

q[k_] := Module[{d = IntegerDigits[k]}, Divisible[k * Times @@ d, Plus @@ d]]; Select[Range[140], q] (* Amiram Eldar, Mar 03 2025 *)

isok(k) = my(d=digits(k)); !((k*vecprod(d)) % vecsum(d)); \\ Michel Marcus, Mar 03 2025

A319507 Smallest number of multiplicative-additive divisors persistence n.

Original entry on oeis.org

1, 2, 36, 3489, 24778899, 566677899999, 47777778999999999999

Offset: 0

Pieter Post, Sep 21 2018

To compute the "multiplicative-additive divisors persistence" of a number, we proceed as follows. Form the product of the digits of the number (A007954) divided by the sum of the digits (A007953). Repeat this process until you reach 0 or 1. If we reach a non-integer, we write 0. The "multiplicative-additive divisors persistence" is the number of steps to reach 0 or 1.

For instance: the multiplicative-additive divisors persistence of 874 is 1, because 874 -> 8 * 7 * 4 / (8 + 7 + 4) = 224/19. This is not an integer, so the process stops after one step.

			The multiplicative additive divisors persistence of 24778899 is 4: 24778899 -> (2032128/54=) 37632 -> (756/21=) 36 -> (18/9=) 2 -> (2/2=) 1.

Cf. A038367, A126789, A003001, A006050, A007953, A007954, A031346.

Offset set to 0. - R. J. Mathar, Jun 30 2020

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A062045 Positive numbers whose product of digits is 12 times their sum.

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A061013 Numbers k such that (product of digits of k) is divisible by (sum of digits of k), where 0's are not permitted.

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A274124 Numbers n such that (product of digits of n) is divisible by (sum of digits of n) and digits of n are in nondecreasing order.

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A055931 Product of the digits of n divides the sum of the digits of n.

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A138566 Numbers n whose digits satisfy the following Diophantine equation: P = (X1* ... *Xr)/(X1+ ... +Xr), where P = prime number, Xi = digits of n.

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A240520 Numbers that set a new integer record for the ratio between the product and the sum of their digits.

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A371631 Primes whose product of nonzero digits divided by the sum of its digits is also prime.

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A381631 Numbers k such that the product of k and its digits is divisible by the sum of its digits.

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