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

0,2

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

Formula

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

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 *)

Crossrefs

Cf. A115334, A210504.

Sequence in context: A285920 A293581 A210047 * A285856 A202257 A278154

Adjacent sequences: A210502 A210503 A210504 * A210506 A210507 A210508

Keyword

nonn

Author

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

Status

approved