A141548 Numbers n whose deficiency is 6.

7, 15, 52, 315, 592, 1155, 2102272, 815634435

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

Keywords

• nonn
• more
• hard

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^(k-1)*(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/3-1. 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

Examples

			a(1) = 7, since 2*7 - sigma(7) = 14 - 8 = 6. - _Timothy L. Tiffin_, Sep 13 2016
		

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.

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).

Programs

Magma

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -6]; // Vincenzo Librandi, Sep 14 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 *)

PARI

is(n)=sigma(n)==2*n-6 \\ Charles R Greathouse IV, Apr 23 2015, corrected by M. F. Hasler, Jul 18 2016

Extensions

a(8) from Donovan Johnson, Dec 08 2011