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).

30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 186, 198, 210, 222, 240, 246, 258, 270, 282, 294, 300, 318, 330, 336, 354, 366, 378, 390, 402, 420, 426, 438, 450, 462, 474, 480, 498, 510, 534, 546, 570, 582, 606, 618, 630

Offset: 1

Author

Amiram Eldar and Paolo P. Lava, May 14 2019

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Crossrefs

Cf. A000203, A005101, A007947, A057723, A307888, A307958, A308029, A308135.

Programs

Maple

with(numtheory): P:=proc(k) local a,n; a:=mul(n,n=factorset(k));
if sigma(k)-a*sigma(k/a)>k then k; fi;  end: seq(P(i),i=1..630);

Mathematica

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]

PARI

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = my(r=rad(n)); sumdiv(n, d, if (rad(d)!=r, d));
isok(n) = s(n) > n; \\ Michel Marcus, May 14 2019