A334679 -id:A334679 - cp's OEIS frontend

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

Scott R. Shannon, May 11 2020

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

			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.

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

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

Subsequence of A052382.

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 *)

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

A334803 Numbers k such that k*p is divisible by k+p and k-p, where k > p > 0 and p = A007954(k) = the product of digits of k.

Original entry on oeis.org

24, 36, 3276, 1886976

Offset: 1

Scott R. Shannon, May 12 2020

If a(5) exists it is at least 3*10^12.

a(5) > 1.5*10^14, if it exists. - Giovanni Resta, May 12 2020

			24 is a term as p = 2*4 = 8 and 24*8 = 192 is divisible by both 24-8 = 16 and 24+8 = 32.
36 is a term as p = 3*6 = 18 and 38*18 = 648 is divisible by both 36-18 = 18 and 36+18 = 54.
3276 is a term as p = 3*2*7*6 = 252 and 3276*252 = 825552 is divisible by both 3276-252 = 3024 and 3276+252 = 3528.
1886976 is a term as p = 1*8*8*6*9*7*6 = 145152 and 1886976*145152 = 273898340352 is divisible by both 1886976-145152 = 1741824 and 1886976+145152 = 2032128.

Cf. A007954, A049102, A085124.

Subsequence of A052382. Intersection of A334679 and A330880.

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

cp's OEIS Frontend

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.

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