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Search: id:A127724
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| A127724 |
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Numbers n that are k-imperfect. |
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+0
4
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| 1, 2, 6, 12, 40, 120, 126, 252, 880, 2520, 2640, 10880, 30240, 32640, 37800, 37926, 55440, 75852, 685440, 758520, 831600, 2600640, 5533920, 6917400, 9102240, 10281600, 11377800, 16687440, 152182800, 206317440, 250311600, 475917120
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OFFSET
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1,2
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COMMENT
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For prime powers p^e, define a multiplicative function rho(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e. A number n is called k-imperfect if there is an integer k such that n=k*rho(n). Sequence A061020 gives a signed version of the rho function. As with multiperfect numbers (A007691), 2-imperfect numbers are also called imperfect numbers. No k-imperfect numbers are known for k>3. As shown by Iannucci, when rho(n) is prime, there is sometimes a technique for generating larger imperfect numbers.
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REFERENCES
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R. K. Guy, Unsolved Problems in Theory of Numbers, Springer, 1994, B1.
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LINKS
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Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.
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MATHEMATICA
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f[p_, e_]:=Sum[(-1)^(e-k) p^k, {k, 0, e}]; rho[n_]:=Times@@(f@@@FactorInteger[n]); Select[Range[10^6], Mod[ #, rho[ # ]]==0&]
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CROSSREFS
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Cf. A127725 (2-imperfect numbers), A127726 (3-imperfect numbers), A127727 (related primes).
Sequence in context: A123045 A094261 A080497 this_sequence A056744 A083001 A119862
Adjacent sequences: A127721 A127722 A127723 this_sequence A127725 A127726 A127727
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KEYWORD
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nice,nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jan 25 2007
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