cp's OEIS Frontend

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

%I A381631 #96 Mar 07 2025 06:32:29
%S A381631 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,
%T A381631 54,60,62,63,66,70,72,80,81,84,88,90,100,101,102,103,104,105,106,107,
%U A381631 108,109,110,111,112,114,116,117,120,123,126,130,132,133,134,135,138,140
%N A381631 Numbers k such that the product of k and its digits is divisible by the sum of its digits.
%C A381631 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.
%C A381631 Terms with a 0 digit form various runs of consecutive terms, such as from 100...00 through to 111...10.
%C A381631 Terms without a 0 digit can form runs of 9 terms: see A381697.
%C A381631 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).
%e A381631 36 is a term because 36*3*6 is divisible by 3+6.
%e A381631 140 is a term because 140*1*4*0 equals 0, which is trivially divisible by 1+4+0.
%t A381631 q[k_] := Module[{d = IntegerDigits[k]}, Divisible[k * Times @@ d, Plus @@ d]]; Select[Range[140], q] (* _Amiram Eldar_, Mar 03 2025 *)
%o A381631 (PARI) isok(k) = my(d=digits(k)); !((k*vecprod(d)) % vecsum(d)); \\ _Michel Marcus_, Mar 03 2025
%Y A381631 Cf. A007953, A098736, A381697.
%Y A381631 Cf. A011540, A038367, A005349.
%K A381631 nonn,base
%O A381631 1,2
%A A381631 _Jakub Buczak_, Mar 02 2025