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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 PROG (PARI) isok(n) = sumdiv(n, d, if ((dv, 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, n), print1(n", "))) \\ 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.)