A349321 Numbers k such that k-1, k+1, 2k-1, 2k+1, 3k-1, 3k+1, 4k-1, 4k+1, 5k-1, and 5k+1 are all primes.

100803789240, 441913177860, 1768738337520, 3906037699410, 5988326187690, 6266477200830, 6905441609220, 6973884137220, 14323608903450, 17683172090430, 20047266723330, 23371434572640, 27904703386560, 29484744885750, 31141493827290, 33202639844220, 34645262968470

Offset: 1

Author

Jon E. Schoenfield, Nov 14 2021

Keywords

• nonn

Comments

All terms are multiples of 2*3*5*7*11 = 2310.

From Jon E. Schoenfield, Mar 21 2022: (Start)

Each term is congruent to one of only

1 residue modulo 2*3*5*7*11 = 2310 (0.04329%),

3 residues modulo 2*3*5*7*11*13 = 30030 (0.00999%),

21 residues modulo 2*3*5*7*...*17 = 510510 (0.00411%),

189 residues modulo 2*3*5*7*...*19 = 9699690 (0.00195%),

2457 residues modulo 2*3*5*7*...*23 = 223092870 (0.00110%),

46683 residues modulo 2*3*5*7*...*29 = 6469693230 (0.00072%),

980343 residues modulo 2*3*5*7*...*31 = 200560490130 (0.00049%), etc.;

making use of these can allow more efficient searching for terms of the sequence.

The Magma program (see Links) generates a list of the possible residues modulo 2*3*5*7*...*31 and tests only numbers having one of those residues. (Note that the program, when run on the Online Magma Calculator, generates only the first three terms of the sequence before being terminated on reaching the 120-second time limit.) (End)

Links

• Jon E. Schoenfield, Magma program

Crossrefs

Cf. A014574, A066388, A118859, A118860, A348348.

Programs

PARI

is_ok(k)=for(j=1,5, if(!isprime(j*k-1), return(0)); if(!isprime(j*k+1), return(0));); return(1); \\ Joerg Arndt, Nov 15 2021