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%I #13 May 15 2020 06:43:21
%S 2,4,6,10,12
%N Numbers k such that k+j is prime for every j, where 1 <= j < k and gcd(j,k) = 1.
%C It was conjectured by Recamán Santos in 1976 and proved by Hausman and Shapiro in 1978 that 12 is the largest k possible.
%C Pomerance & Penney (1977) reported in a letter that they have proved that the conjecture is true. - _Amiram Eldar_, May 15 2020
%D Paulo Ribenboim, The New Book of Prime Number Records, Third ed., Springer-Verlag New York, 1996, p. 42.
%H Miriam Hausman and Harold N. Shapiro, <a href="https://www.jstor.org/stable/2690248">Adding totitives</a>, Mathematics Magazine, Vol. 51, No. 5 (1978), pp. 284-288.
%H Carl Pomerance and David E. Penney, <a href="https://www.jstor.org/stable/2689739">Santos' conjecture</a>, News & Letters, Mathematics Magazine, Vol. 50, No. 2 (1977), p. 107.
%H Bernardo Recamán Santos, <a href="https://www.jstor.org/stable/2689451">Twelve and its totatives</a>, Mathematics Magazine, Vol. 49, No. 5 (1976), pp. 239-240.
%e 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.
%t sQ[n_/;n>1]:=AllTrue[n+Select[Range[n-1],GCD[#,n]==1&],PrimeQ];Select[Range[12],sQ]
%K nonn,fini,full
%O 1,1
%A _Ivan N. Ianakiev_, Jul 26 2019
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