A277691 Smallest k such that (k+i)*prime(n)# - 1 is prime for i = 0, 1, 2, 3, 4 with prime(n)# = A002110(n) the n-th primorial, n>1.

1, 12, 710, 267, 159, 164, 76, 90, 16285, 2168, 3832, 7773, 29849, 34731, 1496, 23485, 51130, 17658, 38908, 38639, 270845, 450432, 57050, 145850, 631632, 240947, 398108, 306349, 288481, 410531, 1516421, 2621329, 781173, 333140, 2997665, 948049, 593835, 1506645, 1216039

Offset

2,2

Comments

5, 11, 17, 23, 29 is the smallest set of 5 primes in arithmetic progression, and may be written (1+i)*3#-1 for i=0 to 4.

Conjecture: for all n>1, there exists an integer k to get 5 primes in arithmetic progression starting with k*prime(n)# - 1.

Links

Table of n, a(n) for n=2..40.

Pierre CAMI, PFGW Script

Example

12*30-1=359 prime, (12+1)*30-1=389 prime, (12+2)*30-1=419 prime, (12+3)*30-1=449 prime, (12+4)*30-1=479 prime, as 30 = 2*3*5 = prime(3)#, so a(2)=12.

Mathematica

Table[Function[p, k = 1; While[Times @@ Boole@ Map[PrimeQ[p (k + #) - 1] &, Range[0, 4]] == 0, k++]; k]@ Product[Prime@ j, {j, n}], {n, 2, 17}] (* or *)
Do[Function[p, k = 1; While[Times @@ Boole@ Map[PrimeQ[p (k + #) - 1] &, Range[0, 4]] == 0, k++]; Print@ k]@ Product[Prime@ j, {j, n}], {n, 2, 23}] (* Michael De Vlieger, Oct 27 2016 *)

Crossrefs

Cf: A002110.

Sequence in context: A203307 A171105 A215686 * A262383 A002196 A141421

Adjacent sequences: A277688 A277689 A277690 * A277692 A277693 A277694

Keyword

nonn

Author

Pierre CAMI, Oct 27 2016

Status

approved