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A066218
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Numbers n such that sigma(n) = sum_{k divides n, k < n} sigma(k).
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17
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OFFSET
| 1,1
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COMMENTS
| I propose this generalization of perfect numbers: for an arithmetical function f, the "f-perfect numbers" are the n such that f(n) = sum of f(k) where k ranges over proper divisors of n. The usual perfect numbers are i-perfect numbers, where i is the identity function. This sequence lists the sigma-perfect numbers. It is not hard to see that the EulerPhi-perfect numbers are the powers of 2 and the d-perfect numbers are the squares of primes (d(n) denotes the number of divisors of n).
Problems: Find an expression generating sigma-perfect numbers. Are there infinitely many of these? Find other interesting sets generated by other f's. 3.
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LINKS
| J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.
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FORMULA
| Integer n = p1^k1 * p2^k2 * ... * pm^km is in this sequence if and only if g(p1^k1)*g(p2^k2)*...*g(pm^km)=2, where g(p^k)=(p^(k+2)-(k+2)*p+k+1)/(p^(k+1)-1)/(p-1) for prime p and integer k>=1. [From Max Alekseyev (maxale(AT)gmail.com), Oct 23 2008]
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EXAMPLE
| Proper divisors of 198 = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99}; sum of their sigma values = 1 + 3 + 4 + 12 + 13 + 12 + 39 + 36 + 48 + 144 + 156 = 468 = sigma(198).
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MATHEMATICA
| f[ x_ ] := DivisorSigma[ 1, x ]; Select[ Range[ 1, 10^5 ], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]
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CROSSREFS
| Sequence in context: A075293 A083264 A202526 * A158222 A156771 A065697
Adjacent sequences: A066215 A066216 A066217 * A066219 A066220 A066221
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KEYWORD
| nonn,more
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AUTHOR
| Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Dec 17 2001
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EXTENSIONS
| More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), May 07 2002
2 more terms from Farideh Firoozbakht (mymontain(AT)yahoo.com), Sep 18 2006
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