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

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

Author

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

Keywords

• nonn
• easy
• base

Comments

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

Examples

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

References

• 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.

Links

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

Crossrefs

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

Cf. A274124.

Programs

Maple

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:

Mathematica

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

PARI

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

Python

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

Extensions

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