cp's OEIS Frontend

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

%I A294900 #51 Jun 18 2019 07:23:59
%S A294900 6,24,28,126,496,8128,5594428,33550336,8589869056,17589794838,
%T A294900 35439846824,49380301744
%N A294900 Numbers k such that k = sum of nonabundant proper divisors of k (A294888).
%C A294900 Naturally, all the terms of A000396, including 137438691328, are in this sequence. - _Antti Karttunen_, Dec 01 2017
%C A294900 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 A294900 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 A294900 Questions from _Iain Fox_, Dec 07 2017: (Start)
%C A294900 Are there an infinite number of abundant terms?
%C A294900 Are all abundant terms in this sequence even?
%C A294900 (End)
%C A294900 No other terms up to 10^10. - _Iain Fox_, Dec 07 2017
%C A294900 a(13) > 6*10^10. - _Giovanni Resta_, Dec 11 2017
%C A294900 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 A294900 From _Iain Fox_, Dec 11 2017: (Start)
%C A294900 The first abundant term without a perfect divisor is 35439846824.
%C A294900 This term and any other abundant terms without perfect divisors are also terms in A125310.
%C A294900 (End)
%H A294900 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%o A294900 (PARI) isok(n) = sumdiv(n, d, if ((d<n) && (sigma(d)<=(2*d)), d)) == n; \\ _Michel Marcus_, Nov 17 2017
%o A294900 (PARI) normalize(f)=f=select(v->v[2],f~)~;if(vecmax(matsize(f)),f,factor(1));
%o A294900 is(n,f=factor(n))=
%o A294900 {
%o A294900 my(p=Mat(f[,1]),g,s);
%o A294900 forvec(v=apply(k->[0,k],f[,2]~),
%o A294900 g=normalize(concat(p,v~));
%o A294900 if(sigma(g,-1)<=2,
%o A294900 s+=factorback(g)
%o A294900 );
%o A294900 );
%o A294900 s==if(sigma(f,-1)>2,n,2*n);
%o A294900 }
%o A294900 forfactored(n=6,10^9, if(is(n[1],n[2]), print1(n[1]", "))) \\ _Charles R Greathouse IV_, Dec 08 2017
%Y A294900 Fixed points of A294888.
%Y A294900 Subsequence of A005835; A000396 is a subsequence.
%Y A294900 Cf. A125310.
%K A294900 hard,nonn,more
%O A294900 1,1
%A A294900 _Antti Karttunen_, Nov 14 2017
%E A294900 a(9) from _Iain Fox_, Dec 07 2017
%E A294900 a(10)-a(12) from _Giovanni Resta_, Dec 11 2017