A309374 Numbers k such that k+j is prime for every j, where 1 <= j < k and gcd(j,k) = 1.

2, 4, 6, 10, 12

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., Springer-Verlag New York, 1996, p. 42.

Links

Table of n, a(n) for n=1..5.

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.

Bernardo Recamán Santos, Twelve and its totatives, Mathematics Magazine, Vol. 49, No. 5 (1976), pp. 239-240.

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[n-1], GCD[#, n]==1&], PrimeQ]; Select[Range[12], sQ]

Crossrefs

Sequence in context: A067852 A253968 A102025 * A226295 A090127 A057910

Adjacent sequences: A309371 A309372 A309373 * A309375 A309376 A309377

Keyword

nonn,fini,full

Author

Ivan N. Ianakiev, Jul 26 2019

Status

approved