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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.
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%I #44 Nov 04 2023 21:52:17

%S 24,36,45,48,144,384,624,672,798,816,3276,3648,4864,5994,7965,18816,

%T 56175,83232,98496,177184,199584,275772,344736,377496,784896,879984,

%U 1372896,1378944,1635795,1886976,2472736,3364416,4575375,6595992,9289728,9377424,28348416,33247872,36387792,58677696

%N 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.

%C 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

%H David A. Corneth, <a href="/A330880/b330880.txt">Table of n, a(n) for n = 1..152</a> (terms <= 10^22; first 82 terms from Giovanni Resta)

%e 24 is a term as p = 2*4 = 8 and 24*8 = 192 is divisible by 24 - 8 = 16.

%e 3648 is a term as p = 3*6*4*8 = 576 and 3648*576 = 2101248 is divisible by 3648-576 = 3072.

%e 1372896 is a term as p = 1*3*7*2*8*9*6 = 18144 and 1372896*18144 = 24909825024 is divisible by 1372896 - 18144 = 1354752.

%t 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 *)

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

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

%Y Subsequence of A052382.

%K nonn,base

%O 1,1

%A _Scott R. Shannon_, May 11 2020