A080224 Number of abundant divisors of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 1, 0, 1, 0

Offset: 1

Author

Reinhard Zumkeller, Feb 07 2003

Keywords

• nonn

Comments

Number of divisors d of n with sigma(d)>2*d (sigma = A000203)

a(n)>0 iff n is abundant: a(A005101(n))>0, a(A000396(n))=0 and a(A005100(n))=0; a(A091191(n))=1; a(A091192(n))>1; a(A091193(n))=n and a(m)<>n for m < A091193(n). - Reinhard Zumkeller, Dec 27 2003

Examples

			Divisors of n=24: {1,2,3,4,6,8,12,24}, two of them are abundant: 12=A005101(1) and 24=A005101(4), therefore a(24)=2.
		

Links

• R. Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Abundant Number.

Crossrefs

Cf. A000005, A000203, A005101, A080225, A080226, A187795, A294890, A294929, A294930, A294937.

Cf. also A294904.

Programs

Maple

A080224 := proc(n)
    a := 0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) > 2*d then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A080224(n),n=1..80) ; # R. J. Mathar, Feb 22 2021

Mathematica

Table[Count[Divisors[n],?(DivisorSigma[1,#]>2#&)],{n,110}] (* _Harvey P. Dale, Jun 14 2013 *)

PARI

a(n) = sumdiv(n, d, sigma(d)>2*d)  \\ Michel Marcus, Mar 09 2013

Formula

a(n) + A080225(n) + A080226(n) = A000005(n).

From Antti Karttunen, Nov 14 2017: (Start)

a(n) = Sum_{d|n} A294937(d).

a(n) = A294929(n) + A294937(n).

a(n) = 1 iff A294930(n) = 1.

(End)