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

1,2

Comments

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

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Cf. A274124.

Sequence in context: A108194 A083158 A254956 * A037264 A274124 A045910

Adjacent sequences: A061010 A061011 A061012 * A061014 A061015 A061016

Keyword

nonn,easy,base

Author

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

Extensions

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

Status

approved