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A309374
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Numbers k such that k+j is prime for every j, where 1 <= j < k and gcd(j,k) = 1.
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0
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OFFSET
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1,1
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COMMENTS
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It was conjectured by Recamán Santos in 1976 and proved by Hausman and Shapiro in 1978 that 12 is the largest k possible.
Pomerance & Penney (1977) reported in a letter that they have proved that the conjecture is true. - Amiram Eldar, May 15 2020
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REFERENCES
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Paulo Ribenboim, The New Book of Prime Number Records, Third ed., Springer-Verlag New York, 1996, p. 42.
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LINKS
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Miriam Hausman and Harold N. Shapiro, Adding totitives, Mathematics Magazine, Vol. 51, No. 5 (1978), pp. 284-288.
Carl Pomerance and David E. Penney, Santos' conjecture, News & Letters, Mathematics Magazine, Vol. 50, No. 2 (1977), p. 107.
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EXAMPLE
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For k = 12 the numbers j are {1,5,7,11} and the numbers k+j are {13,17,19,23}, which are all prime.
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MATHEMATICA
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sQ[n_/; n>1]:=AllTrue[n+Select[Range[n-1], GCD[#, n]==1&], PrimeQ]; Select[Range[12], sQ]
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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