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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 (list; graph; refs; listen; history; text; internal format)
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 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

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

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Last modified July 19 12:32 EDT 2019. Contains 325159 sequences. (Running on oeis4.)