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.
Original entry on oeis.org
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
n = pqrs, p<q<r<s, p+q+r+s = ks; n = 6279 = 3*7*13*23, sum = 3+7+13+23 = 2*23
-
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 *)
A071144
Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 5 distinct prime factors and n is squarefree.
Original entry on oeis.org
3570, 8970, 10626, 15015, 16530, 20706, 24738, 24882, 36890, 38130, 44330, 49938, 51051, 52170, 54834, 55986, 59570, 62985, 68370, 73554, 74613, 77330, 79458, 81770, 87290, 91266, 96162, 96866, 103730, 106314, 116466, 123234, 128570, 129426, 129930, 138890
Offset: 1
n = pqrst, p<q<r<s<t, primes, p+q+r+s+t = kt; n = 8970 = 2*3*5*13*23, sum = 46 = 2*23.
-
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], 5]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]
sdpQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==5&&SquareFreeQ[n]&&Mod[Total[ fi],Max[fi]]==0]; Select[Range[150000],sdpQ] (* Harvey P. Dale, May 04 2023 *)
A071145
Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 6 distinct prime factors and n is squarefree.
Original entry on oeis.org
72930, 106590, 190190, 222870, 335478, 397670, 620310, 836418, 844305, 884442, 1008678, 1195670, 1218945, 1247290, 1704794, 1761110, 1799798, 2086238, 2206022, 2328410, 2485830, 2496585, 2517258, 2863718, 2903538, 3024021, 3157665, 3172785, 3291890
Offset: 1
n = pqrstw, p<q<r<s<t<w, primes, p+q+r+s+t+w = kt; n = 106590 = 2*3*5*11*17*19; sum = 2+3+5+11+17+19 = 57 = 3*19 (quotient=3) (Corrected Mar 06 2006.)
-
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], 6]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]