A294900 Numbers k such that k = sum of nonabundant proper divisors of k (A294888).

6, 24, 28, 126, 496, 8128, 5594428, 33550336, 8589869056, 17589794838, 35439846824, 49380301744

Offset: 1

Author

Antti Karttunen, Nov 14 2017

Keywords

• hard
• nonn
• more

Comments

Naturally, all the terms of A000396, including 137438691328, are in this sequence. - Antti Karttunen, Dec 01 2017

Thus, if there are infinitely many Mersenne primes, then this sequence is also, by definition of even perfect numbers, infinite. - Iain Fox, Dec 02 2017

All non-perfect terms are abundant. Proof: Assume d is a deficient number in this sequence. Because multiples of abundant numbers are abundant, d cannot have an abundant divisor, thus all its divisors are nonabundant. Since d is in this sequence, the sum of its proper divisors, which are all nonabundant, must equal d. However, if this were true, then d would be perfect. Therefore, this sequence contains no deficient numbers. - Iain Fox, Dec 07 2017

Questions from Iain Fox, Dec 07 2017: (Start)

Are there an infinite number of abundant terms?

Are all abundant terms in this sequence even?

(End)

No other terms up to 10^10. - Iain Fox, Dec 07 2017

a(13) > 6*10^10. - Giovanni Resta, Dec 11 2017

In comparison, the numbers which are the sum of their abundant proper divisors seems to be scarcer: up to 6*10^10 only 19514300 and 16333377500 have this property. - Giovanni Resta, Dec 11 2017

From Iain Fox, Dec 11 2017: (Start)

The first abundant term without a perfect divisor is 35439846824.

This term and any other abundant terms without perfect divisors are also terms in A125310.

(End)

Links

• Index entries for sequences where any odd perfect numbers must occur

Crossrefs

Fixed points of A294888.

Subsequence of A005835; A000396 is a subsequence.

Cf. A125310.

Programs

PARI

isok(n) = sumdiv(n, d, if ((dMichel Marcus, Nov 17 2017

PARI

normalize(f)=f=select(v->v[2],f~)~;if(vecmax(matsize(f)),f,factor(1));
is(n,f=factor(n))=
{
my(p=Mat(f[,1]),g,s);
forvec(v=apply(k->[0,k],f[,2]~),
g=normalize(concat(p,v~));
if(sigma(g,-1)<=2,
s+=factorback(g)
);
);
s==if(sigma(f,-1)>2,n,2*n);
}
forfactored(n=6,10^9, if(is(n[1],n[2]), print1(n[1]", "))) \\ Charles R Greathouse IV, Dec 08 2017

Extensions

a(9) from Iain Fox, Dec 07 2017

a(10)-a(12) from Giovanni Resta, Dec 11 2017