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.

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

Cf. A099634, A099635, A099636.

Sequence in context: A192322 A015708 A114822 * A274348 A143758 A101260

Adjacent sequences: A099634 A099635 A099636 * A099638 A099639 A099640

Keyword

nonn

Author

Labos Elemer, Oct 28 2004

Extensions

More terms from Robert G. Wilson v, Nov 04 2004

Status

approved