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A075442
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Slowest-growing sequence of primes whose reciprocals sum to 1.
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4
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2, 3, 7, 43, 1811, 654149, 27082315109, 153694141992520880899, 337110658273917297268061074384231117039, 8424197597064114319193772925959967322398440121059128471513803869133407474043
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| This sequence was mentioned by K. S. Brown. The sequence is generated by a greedy algorithm given by the Mathematica program. The sum converges quadratically.
It is easily shown that this sequence is infinite. For suppose there was a finite representation of unity as a sum of unit fractions with distinct prime denominators. Multiply the equation by the product of all denominators to obtain this product of prime numbers on one side of the equation and a sum of products consisting of this product with always exactly one of the prime numbers removed on the other side. Then each of the prime numbers divides one side of the equation but not the other, since it divides all the products added except exactly one. Contradiction. - Peter C. Heinig (algorithms(AT)gmx.de), Sep 22 2006
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REFERENCES
| R. K. Guy, Unsolved Problems in Number Theory, D11.
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LINKS
| Robert G. Wilson v, Table of n, a(n) for n = 1..14 . [From Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 28 2010]
K. S. Brown, Odd, Greedy and Stubborn (Unit Fractions)
Eric Weisstein's World of Mathematics, Egyptian Fraction
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MATHEMATICA
| x=1; lst={}; Do[n=Ceiling[1/x]; If[PrimeQ[n], n++ ]; While[ !PrimeQ[n], n++ ]; x=x-1/n; AppendTo[lst, n], {10}]; lst
a[n_] := a[n] = Block[{sm = Sum[1/(a[i]), {i, n - 1}]}, NextPrime[ Max[ a[n - 1], 1/(1 - sm)]]]; a[0] = 1; Array[a, 10] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 28 2010]
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CROSSREFS
| Cf. A000058.
Sequence in context: A072713 A129871 A000058 * A082993 A071580 A014546
Adjacent sequences: A075439 A075440 A075441 * A075443 A075444 A075445
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KEYWORD
| nice,nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Sep 16 2002
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