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A099637
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.
1
84, 132, 168, 228, 234, 252, 260, 264, 276, 308, 336, 340, 372, 396, 456, 468, 504, 516, 520, 528, 532, 552, 558, 564, 580, 588, 616, 644, 672, 680, 684, 702, 708, 740, 744, 756, 792, 804, 820, 828, 836, 852, 855, 868, 884, 912, 936, 948, 996, 1008, 1012, 1032
OFFSET
1,1
COMMENTS
Of the first million integers, 75811 (of which 6300 are odd) belong to this sequence. - Robert G. Wilson v, Nov 04 2004
All terms have at least 3 distinct prime factors, and at least 4 prime factors counted with multiplicity. - Robert Israel, Aug 05 2024
LINKS
EXAMPLE
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.
MAPLE
filter:= proc(n) local F, s, p, t;
F:= numtheory:-factorset(n);
s:= convert(F, `+`);
p:= convert(F, `*`);
igcd(s, n) <> igcd(s, p)
end proc:
select(filter, [$1..2000]); # Robert Israel, Aug 05 2024
MATHEMATICA
<<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}]
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Oct 28 2004
EXTENSIONS
More terms from Robert G. Wilson v, Nov 04 2004
STATUS
approved