

A141548


Numbers n whose deficiency is 6.


17




OFFSET

1,1


COMMENTS

a(9) > 10^12.  Donovan Johnson, Dec 08 2011
a(9) > 10^13.  Giovanni Resta, Mar 29 2013
a(9) > 10^18.  Hiroaki Yamanouchi, Aug 21 2018
For all k in A059242, the number m = 2^(k1)*(2^k+5) is in this sequence. This yields further terms 2^46*(2^47+5), 2^52*(2^53+5), 2^140*(2^141+5), ... All even terms known so far and the initial 7 = 2^0*(2^1+5) are of this form. All odd terms beyond a(2) are of the form a(n) = a(k)*p*q, k < n. We have proved that there is no further term of this form with the a(k) given so far.  M. F. Hasler, Apr 23 2015
A term n of this sequence multiplied by a prime p not dividing it is abundant if and only if p < sigma(n)/6 = n/31. For the even terms 592 and 2102272, there is such a prime near this limit (191 resp. 693571) such that n*p is a primitive weird number, cf. A002975. For a(3)=52, the largest such prime, 11, is already too small. Odd weird numbers do not exist within these limits.  M. F. Hasler, Jul 19 2016
Any term x of this sequence can be combined with any term y of A087167 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.  Timothy L. Tiffin, Sep 13 2016


LINKS

Table of n, a(n) for n=1..8.
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), 153163, arXiv:1803.00324 [math.NT], 2018.


EXAMPLE

a(1) = 7, since 2*7  sigma(7) = 14  8 = 6.  Timothy L. Tiffin, Sep 13 2016


MATHEMATICA

lst={}; Do[If[n==Plus@@Divisors[n]n+6, AppendTo[lst, n]], {n, 10^4}]; Print[lst];
Select[Range[1, 10^8], DivisorSigma[1, #]  2 # ==  6 &] (* Vincenzo Librandi, Sep 14 2016 *)


PROG

(PARI) is(n)=sigma(n)==2*n6 \\ Charles R Greathouse IV, Apr 23 2015, corrected by M. F. Hasler, Jul 18 2016
(MAGMA) [n: n in [1..9*10^6]  (SumOfDivisors(n)2*n) eq 6]; // Vincenzo Librandi, Sep 14 2016


CROSSREFS

Cf. A087485 (odd terms).
Cf. A000203, A033880, A005100; A191363 (deficiency 2), A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A101223 (deficiency 10), A141549 (deficiency 12), A141550 (deficiency 14), A125248 (deficiency 16), A223608 (deficiency 18), A223607 (deficiency 20).
Cf. A087167 (abundance 6).
Sequence in context: A042287 A145978 A037376 * A146159 A187986 A039789
Adjacent sequences: A141545 A141546 A141547 * A141549 A141550 A141551


KEYWORD

nonn,more,hard


AUTHOR

Vladimir Joseph Stephan Orlovsky, Aug 16 2008


EXTENSIONS

a(8) from Donovan Johnson, Dec 08 2011


STATUS

approved



