

A294900


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


2



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

1,1


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 nonperfect 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

Table of n, a(n) for n=1..12.
Index entries for sequences where any odd perfect numbers must occur


PROG

(PARI) isok(n) = sumdiv(n, d, if ((d<n) && (sigma(d)<=(2*d)), d)) == n; \\ Michel 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


CROSSREFS

Fixed points of A294888.
Subsequence of A005835; A000396 is a subsequence.
Cf. A125310.
Sequence in context: A219362 A226476 A216793 * A064510 A228383 A249667
Adjacent sequences: A294897 A294898 A294899 * A294901 A294902 A294903


KEYWORD

hard,nonn,more


AUTHOR

Antti Karttunen, Nov 14 2017


EXTENSIONS

a(9) from Iain Fox, Dec 07 2017
a(10)a(12) from Giovanni Resta, Dec 11 2017


STATUS

approved



