This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066230 #34 Jul 31 2025 14:32:09 %S A066230 1,12,196,368,1696,30848,437745,2075648,8341504,33452032,34355150848, %T A066230 562949131337728,2305842943715442688,590295809173294678016 %N A066230 f-perfect numbers, where f(m) = m - 1. %C A066230 f-perfect numbers are defined in A066218. %C A066230 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 A066230 If 2^k - 2*k + 1 is prime (i.e. k in A301744), then 2^(k-1)*(2^k - 2*k + 1) is a term. The only known terms not of this form are 1, 196, and 437745. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 31 2005; updated by _Max Alekseyev_, Jul 30 2025 %C A066230 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 %H A066230 J. Pe, <a href="https://vixra.org/abs/2503.0165">On a Generalization of Perfect Numbers</a>, J. Rec. Math., 31(3) (2002-2003), 168-172. %e A066230 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 A066230 g[ n_ ] := DivisorSigma[ 1, n ]-n-DivisorSigma[ 0, n ]+2; For[ n=1, True, n++, If[ g[ n ]==n, Print[ n ] ] ] %Y A066230 Cf. A066218, A066511, A066229, A301744. %K A066230 nonn,more %O A066230 1,2 %A A066230 _Joseph L. Pe_, Dec 18 2001 %E A066230 Edited by _Dean Hickerson_, Jan 10 2002. %E A066230 More terms from _Jason Earls_, May 14 2002 %E A066230 2 more terms from _Farideh Firoozbakht_, Sep 18 2006 %E A066230 a(11) from _Donovan Johnson_, Jun 25 2012 %E A066230 a(12)-a(14) from _Max Alekseyev_, Jul 11 2025