

A330880


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


3



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
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OFFSET

1,1


COMMENTS

Every term m is the sum of two 7smooth numbers. Proof: Since (mp)  m*p, we have m*p = (m  p)*k for some k > 0. Suppose m is not the sum of two 7smooth numbers. Then m  p is not 7smooth 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 (first 82 terms from Giovanni Resta, terms <= 10^22)


EXAMPLE

24 is a term as p = 2*4 = 8 and 24*8 = 192 is divisible by 248 = 16.
3648 is a term as p = 3*6*4*8 = 576 and 3648*576 = 2101248 is divisible by 3648576 = 3072.
1372896 is a term as p = 1*3*7*2*8*9*6 = 18144 and 1372896*18144 = 24909825024 is divisible by 137289618144 = 1354752.


MATHEMATICA

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


PROG

(PARI) isok(m) = my(p=vecprod(digits(m))); p && (mp) && !((m*p) % (mp)); \\ 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



