A083263 Numbers k such that the difference of the largest and smallest prime factors of k divides k.

6, 12, 18, 24, 30, 36, 48, 54, 60, 70, 72, 90, 96, 108, 120, 140, 144, 150, 162, 180, 192, 198, 210, 216, 240, 270, 280, 286, 288, 300, 324, 350, 360, 384, 396, 420, 432, 450, 480, 486, 490, 510, 540, 560, 572, 576, 594, 600, 630, 646, 648, 700, 720, 750, 768

Offset: 1

Author

Labos Elemer, May 12 2003

Keywords

• nonn

Examples

			Every number k of the form 2^i * 3^j * m is a term because 3 - 2 = 1 is always a divisor of k.
Every number k of the form 2 * p * (p+2) * m is a term if p and p+2 form a twin prime pair.
Other terms include some in which the difference d = gpf(k) - lpf(k) > 2 is prime (e.g., 30 = 2*3*5 = 3*10; d = 5 - 2 = 3) and some in which it is composite (e.g., 8710 = 2*5*13*67 = 65*134; d = 67 - 2 = 65).
All terms are even. - _Jon E. Schoenfield_, Jul 10 2018
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A033845, A071141, A006530, A020639.

Programs

Mathematica

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; mi[x_] := Min[ba[x]] Do[s=ma[ba[n]]-mi[ba[n]]; If[Mod[n, s]==0, Print[{n, ba[n], s}]], {n, 1, 10000}]

Formula

Solutions to x mod (A006530(x) - A020639(x)) = 0.

Extensions

Edited by Jon E. Schoenfield, Jul 10 2018