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 A191363 Numbers n such that sigma(n) = 2*n - 2. 14
 3, 10, 136, 32896, 2147516416 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Let k be a nonnegative integer such that F(k) = 2^(2^k) + 1 is prime (a Fermat prime A019434), then n = (F(k)-1)*F(k)/2 appears in the sequence. Conjecture: a(1)=3 is the only odd member of the sequence. Conjecture: All elements of the sequence are of the above form derived from Fermat primes. The sequence has 5 (known) elements in common with sequences A055708 (n-1 | sigma(n)) and A056006 (n | sigma(n)+2) since a(n) is a subsequence of both. The first five members of the sequence are respectively congruent to 3, 4, 4, 4, 4 modulo 6. There are no further entries after a(5) up 8 * 10^9. Up to n = 1312 * 10^8 there are no further entries in the class congruent to 4 modulo 6. a(6) > 10^12. - Donovan Johnson, Dec 08 2011 a(6) > 10^13. - Giovanni Resta, Mar 29 2013 a(6) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018 See A125246 for numbers with deficiency 4, i.e., sigma(n) = 2*n - 4, and A141548 for numbers with deficiency 6. - M. F. Hasler, Jun 29 2016 and Jul 17 2016 A term n of this sequence multiplied with a prime p not dividing it is abundant if and only if p < n-1. For each of a(2..5) there is such a prime near this limit (here: 7, 127, 30197, 2147483647) such that a(k)*p is a primitive weird number, cf. A002975. - M. F. Hasler, Jul 19 2016 Any term x of this sequence can be combined with any term y of A088831 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. [Proof: If x = a(n) and y = A088831(m), then sigma(x) = 2x-2 and sigma(y) = 2y+2.  Thus, sigma(x)+sigma(y) = (2x-2)+(2y+2) = 2x+2y = 2(x+y), which implies that (sigma(x)+sigma(y))/(x+y) = 2(x+y)/(x+y) = 2.] - Timothy L. Tiffin, Sep 13 2016 At least the first five terms are a subsequence of A295296 and of A295298. - David A. Corneth, Antti Karttunen, Nov 26 2017 Conjectures: all terms are second hexagonal numbers (A014105). There are no terms with middle divisors. - Omar E. Pol, Oct 31 2018 LINKS Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018. FORMULA a(n) = (A019434(n)-1)*A019434(n)/2 for all terms known so far. - M. F. Hasler, Jun 29 2016 EXAMPLE For n=1, a(1) = 3 since sigma(3) = 4 = 2*3 -2 MATHEMATICA ok[n_] := DivisorSigma[1, n] == 2*n-2; Select[ Table[ 2^(2^k-1) * (2^(2^k)+1), {k, 0, 5}], ok] (* Jean-François Alcover, Sep 14 2011, after conjecture *) Select[Range[10^6], DivisorSigma[1, #] == 2 # - 2 &] (* Michael De Vlieger, Sep 14 2016 *) PROG (PARI) zp(a, b) = {my(c, c1, s); c = a; c1 = 2*c-2; while(c

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Last modified October 17 12:56 EDT 2019. Contains 328112 sequences. (Running on oeis4.)