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%I #49 Jan 22 2023 20:50:29
%S 24,30,42,54,66,78,102,114,138,174,186,222,246,258,282,304,318,354,
%T 366,402,426,438,474,498,534,582,606,618,642,654,678,762,786,822,834,
%U 894,906,942,978,1002,1038,1074,1086,1146,1158,1182,1194,1266,1338,1362
%N Numbers k whose abundance is 12: sigma(k) - 2*k = 12.
%C Numbers k such that sigma(k) = 2k + 12. - _Wesley Ivan Hurt_, Jul 11 2013
%C 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
%C From _Tomohiro Yamada_, Jan 01 2023: (Start)
%C 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.
%C Problem: is 54 the only term of this sequence which is of neither type given above? (End)
%H Donovan Johnson, <a href="/A141545/b141545.txt">Table of n, a(n) for n = 1..10000</a>
%H Farideh Firoozbakht and M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1.
%e 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
%t lst={};Do[If[n==Plus@@Divisors[n]-n-12,AppendTo[lst,n]],{n,10^4}];Print[lst];
%t Select[Range[1, 10^4], DivisorSigma[1, #] - 2 # == 12 &] (* _Vincenzo Librandi_, Sep 14 2016 *)
%o (Magma) [n: n in [1..1400] | (SumOfDivisors(n)-2*n) eq 12]; // _Vincenzo Librandi_, Sep 14 2016
%o (PARI) is(n)=sigma(n)==2*n+12 \\ _Charles R Greathouse IV_, Feb 21 2017
%Y Cf. A000203, A005101, A141549 (deficiency 12).
%Y Cf. A076496 (sigma(k) - a*k = 12).
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Aug 16 2008
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