A099637 - OEIS

%I #17 Aug 05 2024 15:27:48

%S 84,132,168,228,234,252,260,264,276,308,336,340,372,396,456,468,504,

%T 516,520,528,532,552,558,564,580,588,616,644,672,680,684,702,708,740,

%U 744,756,792,804,820,828,836,852,855,868,884,912,936,948,996,1008,1012,1032

%N Numbers such that gcd(Sum,n) = A099635 and gcd(Sum,Product) = A099636 are not identical. Sum and Product here are the sum and product of all distinct prime factors of n.

%C Of the first million integers, 75811 (of which 6300 are odd) belong to this sequence. - _Robert G. Wilson v_, Nov 04 2004

%C All terms have at least 3 distinct prime factors, and at least 4 prime factors counted with multiplicity. - _Robert Israel_, Aug 05 2024

%H Robert Israel, <a href="/A099637/b099637.txt">Table of n, a(n) for n = 1..10000</a>

%e 84 is here because its factor list = {2,3,7} and sum = 2 + 3 + 7 = 12, product = 2*3*7 = 42, gcd(12,84) = 12, gcd(12,42) = 6 != 12.

%p filter:= proc(n) local F,s,p,t;

%p   F:= numtheory:-factorset(n);

%p   s:= convert(F,`+`);

%p   p:= convert(F,`*`);

%p   igcd(s,n) <> igcd(s,p)

%p end proc:

%p select(filter, [$1..2000]); # _Robert Israel_, Aug 05 2024

%t <<NumberTheory`NumberTheoryFunctions` pf[x_] :=PrimeFactorList[x];a=Table[Max[pf[w]], {w, 2, m}]; Table[GCD[Apply[Plus, pf[w]], Apply[Plus, pf[w]]], {w, 1, 100}]

%t PrimeFactors[n_Integer] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; fQ[n_] := Block[{pf = PrimeFactors[n]}, GCD[Plus @@ pf, n] == GCD[Plus @@ pf, Times @@ pf]]; Select[ Range[1039], ! fQ[ # ] &] (* _Robert G. Wilson v_, Nov 04 2004 *)

%Y Cf. A099634, A099635, A099636.

%K nonn

%O 1,1

%A _Labos Elemer_, Oct 28 2004

%E More terms from _Robert G. Wilson v_, Nov 04 2004