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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

Table of n, a(n) for n=1..5.

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-Fran├žois 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<b, s = sigma(c); if(s == c1, print(c); ); c1 = c1 + 2; c = c+1); }

(PARI) a(k)=(2^2^k+1)<<(2^k-1) \\ For k<6. - M. F. Hasler, Jul 27 2016

(MAGMA) [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -2]; // Vincenzo Librandi, Sep 15 2016

CROSSREFS

Cf. A000203, A002975, A056006, A055708, A088831 (abundance 2).

Cf. A033880, A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A125248 (deficiency 16).

Cf. A295296, A295298.

Sequence in context: A173415 A199232 A056006 * A291950 A067999 A256164

Adjacent sequences:  A191360 A191361 A191362 * A191364 A191365 A191366

KEYWORD

nonn,hard,more

AUTHOR

Luis H. Gallardo, May 31 2011

STATUS

approved

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