A210505 Numbers k for which 2*k+7, 4*k+7, 6*k+7, 8*k+7, 10*k+7 and 12*k+7 are primes.

0, 75, 1380, 1725, 4575, 7095, 10020, 10620, 31800, 38355, 58710, 61170, 67125, 92235, 92310, 94845, 118530, 137415, 156000, 168765, 189705, 238815, 249450, 257370, 339375, 353925, 507270, 584265, 590040, 617265, 625845, 631740, 761760, 845295, 866910, 943605

Offset: 0

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 25 2013

Keywords

• nonn

Comments

Conjecture. For every odd prime p there exist infinitely many numbers k for which 2*k+p, 4*k+p, ..., 2*(p-1)*k+p are primes.

For p=3, cf. A115334, for p=5, cf. A210504. This sequence corresponds to p=7.

In general case of prime p, every k == 0 (mod Product{p_2*p_3*...*p_k)), where p_k is the maximal prime < p.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A115334, A210504.

Programs

Mathematica

Select[Range[0, 1000000], PrimeQ[2*# + 7] && PrimeQ[4*# + 7] && PrimeQ[6*# + 7] && PrimeQ[ 8*# + 7] && PrimeQ[ 10*# + 7] && PrimeQ[ 12*# + 7] &] (* T. D. Noe, Jan 25 2013 *)
Select[Range[0,950000],AllTrue[#*Range[2,12,2]+7,PrimeQ]&] (* Harvey P. Dale, Aug 16 2024 *)

Formula

a(n) == 0 (mod 15).