

A309374


Numbers k such that k+j is prime for every j, where 1 <= j < k and gcd(j,k) = 1.


0




OFFSET

1,1


COMMENTS

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


REFERENCES

Paulo Ribenboim, The New Book of Prime Number Records, Third ed., SpringerVerlag New York, 1996, p. 42.


LINKS

Miriam Hausman and Harold N. Shapiro, Adding totitives, Mathematics Magazine, Vol. 51, No. 5 (1978), pp. 284288.
Carl Pomerance and David E. Penney, Santos' conjecture, News & Letters, Mathematics Magazine, Vol. 50, No. 2 (1977), p. 107.


EXAMPLE

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.


MATHEMATICA

sQ[n_/; n>1]:=AllTrue[n+Select[Range[n1], GCD[#, n]==1&], PrimeQ]; Select[Range[12], sQ]


CROSSREFS



KEYWORD

nonn,fini,full


AUTHOR



STATUS

approved



