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A100715
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Pseudoperfect (or semiperfect) k-brilliant numbers for some k>1: some set of proper divisors of a(n) sums to a(n) and a(n) = p(1)p(2)...p(m) for primes all with the same number of digits.
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0
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6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 140, 144, 150, 160, 162, 168, 180, 192, 196, 200, 210, 216, 224, 240, 252
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Since every multiple of a semiperfect number is semiperfect, there are an infinite number of values in this sequence and also an infinite number of values in the complement (pseudoperfect or semiperfect numbers which are not k-brilliant numbers).
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REFERENCES
| Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers." B2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 45-53, 1994.
Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., p. 177, 1992.
Zachariou, A. and Zachariou, E. "Perfect, Semi-Perfect and Ore Numbers." Bull. Soc. Math. Greece (New Ser.) 13, 12-22, 1972.
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LINKS
| Eric Weisstein's World of Mathematics, Semiperfect Number
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FORMULA
| a(n) is an element in the intersection of A005835 and A078972. a(n) in A005835 and a(n) is a semiprime with the same number of digits in each prime factor.
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EXAMPLE
| 6 = 2 * 3 is 2-brilliant.
12 = 2 * 2 * 3 is 3-brilliant.
18 = 2 * 3 * 3 is 3-brilliant.
20 = 2 * 2 * 5 is 3-brilliant.
24 = 2 * 2 * 2 * 3 is 4-brilliant.
28 = 2 * 2 * 7 is 3-brilliant.
30 = 2 * 3 * 5 is 3-brilliant.
36 = 2 * 2 * 3 * 3 is 4-brilliant.
40 = 2 * 2 * 2 * 5 is 4-brilliant.
264 is not in the sequence because it is pseudoperfect but 264 = 2 * 2 * 2 * 3 * 11 and 11 has more digits than 2.
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CROSSREFS
| Cf. A005835, A078972, A001358.
Sequence in context: A204829 A005835 A007620 * A205525 A094519 A088723
Adjacent sequences: A100712 A100713 A100714 * A100716 A100717 A100718
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KEYWORD
| easy,nonn,base
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 11 2004
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