OFFSET
1,1
COMMENTS
Numbers k such that sigma(k) = 2k + 12. - Wesley Ivan Hurt, Jul 11 2013
Any term x = a(m) can be combined with any term y = A141549(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2. Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one. So far, these two sequences have not produced an amicable pair. However, if one is ever found, then it will exhibit y-x = 12. - Timothy L. Tiffin, Sep 13 2016
From Tomohiro Yamada, Jan 01 2023: (Start)
6p belongs to this sequence if p > 3 is prime since sigma(6p) = 12(p + 1) = 12p + 12. Moreover, 2^m * (2^(m+1) - 13) is also a term of this sequence if 2^(m+1) - 13 is prime (m+1 is a term of A096818) since sigma(2^m * (2^(m+1) - 13)) = (2^(m+1) + 1) * (2^(m+1) - 13) = 2^(m+1) * (2^(m+1) - 13) + 12. So 24, 304, 127744, 33501184, and 8589082624 also belong to this sequence.
Problem: is 54 the only term of this sequence which is of neither type given above? (End)
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..10000
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
EXAMPLE
30 is in the sequence since sigma(30) = sigma(2*3*5) = sigma(2)*sigma(3)*sigma(5) = 3*4*6 = 72 = 2(30)+12. Since this is the second such number whose abundance is 12, a(2) = 30. - Wesley Ivan Hurt, Jul 11 2013
MATHEMATICA
lst={}; Do[If[n==Plus@@Divisors[n]-n-12, AppendTo[lst, n]], {n, 10^4}]; Print[lst];
Select[Range[1, 10^4], DivisorSigma[1, #] - 2 # == 12 &] (* Vincenzo Librandi, Sep 14 2016 *)
PROG
(Magma) [n: n in [1..1400] | (SumOfDivisors(n)-2*n) eq 12]; // Vincenzo Librandi, Sep 14 2016
(PARI) is(n)=sigma(n)==2*n+12 \\ Charles R Greathouse IV, Feb 21 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Joseph Stephan Orlovsky, Aug 16 2008
STATUS
approved