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A294900 Numbers k such that k = sum of nonabundant proper divisors of k (A294888). 3

%I #51 Jun 18 2019 07:23:59

%S 6,24,28,126,496,8128,5594428,33550336,8589869056,17589794838,

%T 35439846824,49380301744

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

%C Naturally, all the terms of A000396, including 137438691328, are in this sequence. - _Antti Karttunen_, Dec 01 2017

%C 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

%C 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

%C Questions from _Iain Fox_, Dec 07 2017: (Start)

%C Are there an infinite number of abundant terms?

%C Are all abundant terms in this sequence even?

%C (End)

%C No other terms up to 10^10. - _Iain Fox_, Dec 07 2017

%C a(13) > 6*10^10. - _Giovanni Resta_, Dec 11 2017

%C 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

%C From _Iain Fox_, Dec 11 2017: (Start)

%C The first abundant term without a perfect divisor is 35439846824.

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

%C (End)

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%o (PARI) isok(n) = sumdiv(n, d, if ((d<n) && (sigma(d)<=(2*d)), d)) == n; \\ _Michel Marcus_, Nov 17 2017

%o (PARI) normalize(f)=f=select(v->v[2],f~)~;if(vecmax(matsize(f)),f,factor(1));

%o is(n,f=factor(n))=

%o {

%o my(p=Mat(f[,1]),g,s);

%o forvec(v=apply(k->[0,k],f[,2]~),

%o g=normalize(concat(p,v~));

%o if(sigma(g,-1)<=2,

%o s+=factorback(g)

%o );

%o );

%o s==if(sigma(f,-1)>2,n,2*n);

%o }

%o forfactored(n=6,10^9, if(is(n[1],n[2]), print1(n[1]", "))) \\ _Charles R Greathouse IV_, Dec 08 2017

%Y Fixed points of A294888.

%Y Subsequence of A005835; A000396 is a subsequence.

%Y Cf. A125310.

%K hard,nonn,more

%O 1,1

%A _Antti Karttunen_, Nov 14 2017

%E a(9) from _Iain Fox_, Dec 07 2017

%E a(10)-a(12) from _Giovanni Resta_, Dec 11 2017

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)