A330880 Numbers m such that m*p is divisible by m-p, where m > p > 0 and p = A007954(m) = the product of digits of m.

24, 36, 45, 48, 144, 384, 624, 672, 798, 816, 3276, 3648, 4864, 5994, 7965, 18816, 56175, 83232, 98496, 177184, 199584, 275772, 344736, 377496, 784896, 879984, 1372896, 1378944, 1635795, 1886976, 2472736, 3364416, 4575375, 6595992, 9289728, 9377424, 28348416, 33247872, 36387792, 58677696

Offset

1,1

Comments

Every term m is the sum of two 7-smooth numbers. Proof: Since (m-p) | m*p, we have m*p = (m - p)*k for some k > 0. Suppose m is not the sum of two 7-smooth numbers. Then m - p is not 7-smooth and so there exists a prime q > 7 such that q | (m - p). Since q doesn't divide p and q | (m - p) but (m - p) | m*p, we have q | m. But since q | m and q | (m - p) we have q | (m - (m - p)) = p, a contradiction. Q.e.d. - David A. Corneth, Jun 15 2020

Links

David A. Corneth, Table of n, a(n) for n = 1..152 (terms <= 10^22; first 82 terms from Giovanni Resta)

Example

24 is a term as p = 2*4 = 8 and 24*8 = 192 is divisible by 24 - 8 = 16.
3648 is a term as p = 3*6*4*8 = 576 and 3648*576 = 2101248 is divisible by 3648-576 = 3072.
1372896 is a term as p = 1*3*7*2*8*9*6 = 18144 and 1372896*18144 = 24909825024 is divisible by 1372896 - 18144 = 1354752.

Mathematica

npdQ[n_]:=Module[{p=Times@@IntegerDigits[n]}, n>p>0&&Divisible[n*p, n-p]]; Select[Range[6*10^7], npdQ] (* Harvey P. Dale, Jun 14 2020 *)

Prog

(PARI) isok(m) = my(p=vecprod(digits(m))); p && (m-p) && !((m*p) % (m-p)); \\ Michel Marcus, May 12 2020

Crossrefs

Cf. A334679, A334803, A007954, A049102, A085124.

Subsequence of A052382.

Sequence in context: A067341 A307682 A290016 * A195008 A054775 A273088

Adjacent sequences: A330877 A330878 A330879 * A330881 A330882 A330883

Keyword

nonn,base

Author

Scott R. Shannon, May 11 2020

Status

approved