A071143 - OEIS

%I #17 May 01 2018 14:05:44

%S 3135,6279,8855,10695,11571,16095,17255,17391,20615,20735,26691,28083,

%T 31031,36519,41151,41615,45695,46655,47859,48495,50439,54131,56823,

%U 57239,59295,61295,66215,72611,76055,76479,80135,84135,88595,89999,90951,93651,94611

%N 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.

%H Donovan Johnson, <a href="/A071143/b071143.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%t 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}]

%t 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 *)

%t 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 *)

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

%K nonn

%O 1,1

%A _Labos Elemer_, May 13 2002

%E Definition clarified by _Harvey P. Dale_, May 01 2018