A071143 Numbers n such that (i) the sum of the distinct primes dividing n is divisible by the largest prime dividing n and (ii) n has exactly 4 distinct prime factors and (iii) n is squarefree.

3135, 6279, 8855, 10695, 11571, 16095, 17255, 17391, 20615, 20735, 26691, 28083, 31031, 36519, 41151, 41615, 45695, 46655, 47859, 48495, 50439, 54131, 56823, 57239, 59295, 61295, 66215, 72611, 76055, 76479, 80135, 84135, 88595, 89999, 90951, 93651, 94611

Offset

1,1

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

Formula

A008472(n)/A006530(n) is integer; A001221(n) = 4, n is squarefree.

Example

n = pqrs, p<q<r<s, p+q+r+s = ks; n = 6279 = 3*7*13*23, sum = 3+7+13+23 = 2*23

Mathematica

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 4]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]
s = {}; Do[Length[f=FactorInteger@n] == 4 && Max[(t = Transpose@f)[[2]]] == 1 && Mod[Plus @@ t[[1]], t[[1, -1]]] == 0 && AppendTo[s, n], {n, 3, 10^6, 2}]; s (* 12 times faster, Giovanni Resta, Apr 10 2013 *)
sdpQ[n_]:=Module[{fi=FactorInteger[n][[All, 1]]}, Divisible[Total[fi], Last[ fi]] &&Length[fi]==4&&SquareFreeQ[n]]; Select[Range[100000], sdpQ] (* Harvey P. Dale, May 01 2018 *)

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Sequence in context: A045172 A131276 A045075 * A284984 A061048 A250849

Adjacent sequences: A071140 A071141 A071142 * A071144 A071145 A071146

Keyword

nonn

Author

Labos Elemer, May 13 2002

Extensions

Definition clarified by Harvey P. Dale, May 01 2018

Status

approved