A381697 -id:A381697 - cp's OEIS frontend

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

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

cp's OEIS Frontend

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

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