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%I #20 Dec 15 2017 17:35:46
%S 1,12,196,368,1696,30848,437745,2075648,8341504,33452032,34355150848
%N f-perfect numbers defined by f(n) = n - 1 (where f-perfect numbers are defined in A066218).
%C Equivalently, let g(n) = sigma(n)-n-d(n)+2, where d(n) is the number of divisors of n and sigma(n) is their sum. Then n is in the sequence if g(n)=n.
%C It seems that if 2^(i+1)-(2*i+1) is prime, 2^i*(2^(i+1)-(2*i+1)) is in the list. For example i in {2, 4, 5, 7, 10, 11, 12, 17, 24, 30, 34, 113, 151, 185}. No other exceptions than 1, 196 and 437745 for n < 10^8. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 31 2005
%C If 2^(i + 1)-(2i + 1) is prime then n = 2^i*(2^(i + 1)-(2i + 1)) is in the sequence because sigma(n)-d(n) + 2 = (2^(i + 1)-1)*(2^(i + 1)-2i)-2(i + 1) + 2 = 2^(i + 1)*(2^(i + 1)-(2i + 1)) = 2n, so sigma(n)-n-d(n) + 2 = n. -_Farideh Firoozbakht_, Sep 18 2006
%C a(12) > 2*10^11. - _Donovan Johnson_, Jun 25 2012
%C a(12) > 10^13. - _Giovanni Resta_, Aug 21 2013
%H J. Pe, <a href="http://www.numeratus.net/fperfect/fperfect.html">On a Generalization of Perfect Numbers</a>, J. Rec. Math., 31(3) (2002-2003), 168-172.
%e f(12) = 11 = 0 + 1 + 2 + 3 + 5 = f(1) + f(2) + f(3) + f(4) + f(6), hence 12 is a term of the sequence.
%t g[ n_ ] := DivisorSigma[ 1, n ]-n-DivisorSigma[ 0, n ]+2; For[ n=1, True, n++, If[ g[ n ]==n, Print[ n ] ] ]
%Y Cf. A066218, A066511, A066229.
%K nonn,more
%O 1,2
%A _Joseph L. Pe_, Dec 18 2001
%E Edited by _Dean Hickerson_, Jan 10, 2002.
%E More terms from _Jason Earls_, May 14 2002
%E 2 more terms from _Farideh Firoozbakht_, Sep 18 2006
%E a(11) from _Donovan Johnson_, Jun 25 2012
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