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A308127 Non-coreful abundant numbers: numbers k such that ncsigma(k) > k, where ncsigma(k) is the sum of the non-coreful divisors of k (A308135).

%I A308127 #12 Jul 23 2019 03:25:34
%S A308127 30,42,60,66,70,78,84,90,102,114,120,126,132,138,150,156,168,174,180,
%T A308127 186,198,210,222,240,246,258,270,282,294,300,318,330,336,354,366,378,
%U A308127 390,402,420,426,438,450,462,474,480,498,510,534,546,570,582,606,618,630
%N A308127 Non-coreful abundant numbers: numbers k such that ncsigma(k) > k, where ncsigma(k) is the sum of the non-coreful divisors of k (A308135).
%C A308127 Non-coreful divisor d of a number k is a divisor such that rad(d) != rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
%H A308127 Amiram Eldar, <a href="/A308127/b308127.txt">Table of n, a(n) for n = 1..10000</a>
%H A308127 G. E. Hardy and M. V. Subbarao, <a href="http://oeis.org/A005934/a005934.pdf">Highly powerful numbers</a>, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
%e A308127 60 is in the sequence since its non-coreful divisors are 1, 2, 3, 4, 5, 6, 10, 12, 15, and 20 whose sum is 78 > 60.
%p A308127 with(numtheory): P:=proc(k) local a,n; a:=mul(n,n=factorset(k));
%p A308127 if sigma(k)-a*sigma(k/a)>k then k; fi;  end: seq(P(i),i=1..630);
%t A308127 f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncAbQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]) > n; Select[Range[2, 1000], ncAbQ]
%o A308127 (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o A308127 s(n) = my(r=rad(n)); sumdiv(n, d, if (rad(d)!=r, d));
%o A308127 isok(n) = s(n) > n; \\ _Michel Marcus_, May 14 2019
%Y A308127 Cf. A000203, A005101, A007947, A057723, A307888, A307958, A308029, A308135.
%K A308127 nonn
%O A308127 1,1
%A A308127 _Amiram Eldar_ and _Paolo P. Lava_, May 14 2019