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A383080 Numbers k such that sopf(k) does not divide evenly sopfr(k).

%I A383080 #24 Apr 17 2025 09:37:06
%S A383080 12,18,20,24,28,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,84,88,90,
%T A383080 92,96,98,99,104,108,112,116,117,120,124,126,132,135,136,140,144,147,
%U A383080 148,150,152,153,156,160,162,164,168,171,172,175,176,180,184,188,189,198
%N A383080 Numbers k such that sopf(k) does not divide evenly sopfr(k).
%C A383080 First differs from A059404 and A323055 at n = 59.
%C A383080 a(n) has a square factor, A008683(a(n)) = 0.
%C A383080 If p and q are distinct primes, p*q^k is in the sequence iff p + q does not divide k - 1. - _Robert Israel_, Apr 16 2025
%e A383080 12 is a term because sopf(12)=5 does not evenly divide sopfr(12)=7.
%e A383080 18 is a term because sopf(18)=5 does not evenly divide sopfr(18)=8.
%e A383080 20 is a term because sopf(20)=7 does not evenly divide sopfr(20)=9.
%p A383080 filter:= proc(n) local F,t;
%p A383080   F:= ifactors(n)[2];
%p A383080   add(t[1]*t[2],t=F) mod add(t[1],t=F) <> 0
%p A383080 end proc:
%p A383080 select(filter, [$2..300]); # _Robert Israel_, Apr 16 2025
%t A383080 q[k_] := Module[{f = FactorInteger[k]}, !Divisible[Plus @@ Times @@@ f, Plus @@ f[[;; , 1]]]]; Select[Range[200], q] (* _Amiram Eldar_, Apr 16 2025 *)
%o A383080 (Sage) def spf(k):
%o A383080     fl = list(factor(k))
%o A383080     sr = sum(p * e for p, e in fl)
%o A383080     sd = sum(p for p, _ in fl)
%o A383080     return sd, sr
%o A383080 def output(limit=198):
%o A383080     results = []
%o A383080     for k in range(2, limit + 1):
%o A383080         sd, sr = spf(k)
%o A383080         if 0 < sd and sr % sd != 0:
%o A383080             results.append(k)
%o A383080     return results
%o A383080 print(output())
%o A383080 (PARI) isok(k) = if (k>1, my(f=factor(k)); sum(j=1, #f~, f[j,1]*f[j,2]) % sum(j=1, #f~, f[j,1])); \\ _Michel Marcus_, Apr 16 2025
%Y A383080 Cf. A001414, A008472, A008683.
%Y A383080 Cf. A059404, A323055.
%K A383080 nonn
%O A383080 1,1
%A A383080 _Torlach Rush_, Apr 15 2025