A083263 -id:A083263 - cp's OEIS frontend

A083264 Numbers k such that the difference d of the largest and smallest prime factors of k is a composite divisor of k.

198, 396, 510, 594, 792, 966, 990, 1020, 1188, 1386, 1530, 1566, 1584, 1782, 1932, 1980, 2040, 2178, 2376, 2550, 2590, 2772, 2898, 2970, 3060, 3132, 3168, 3198, 3564, 3570, 3864, 3960, 4080, 4158, 4230, 4356, 4590, 4698, 4752, 4830, 4950, 5100, 5180

Offset: 1

Labos Elemer, May 12 2003

From David A. Corneth, Jul 14 2018: (Start)
No term k is a perfect power (or 1). If k is a perfect power then it's divisible by 0, a contradiction. Hence a term k has at least two prime factors.
All terms are even. Suppose a term k is odd. Then the smallest prime factor is > 2. Since k has at least two prime factors which are odd, the difference between the largest and smallest prime factor is even hence k is even. A contradiction, hence all terms are even.
All terms are of the form 2 * (p - 2) * p * m where p - 2 is composite, p is prime and m has all, if any, of its prime factors between 2 and p (inclusive). (End)

			198 = 2*3*3*11 = 2*9*11 is in the sequence where d = 11 - 2 = 9 is composite.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A025584, A033845, A071141, A006530, A020639, A083263.

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&&Greater[s, 2]&&!PrimeQ[s], Print[n]], {n, 1, 20000}]
dllpfQ[n_]:=Module[{c=Transpose[FactorInteger[n]][[1]],d},d=Last[c]-First[ c];If[d==0,d=1];Divisible[n,d]&&d>2&&CompositeQ[d]]; Select[ Range[ 6000],dllpfQ] (* Harvey P. Dale, Sep 26 2014 *)

isok(n) = if (n>1, my(f=factor(n)[,1], d = vecmax(f) - vecmin(f)); (d > 1) && !isprime(d) && !(n % d)); \\ Michel Marcus, Jul 09 2018

Solutions to x mod d = 0 where d = A006530(x) - A020639(x) is composite.

Name, Formula, and Example simplified by Jon E. Schoenfield, Jul 14 2018

cp's OEIS Frontend

A083264 Numbers k such that the difference d of the largest and smallest prime factors of k is a composite divisor of k.

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