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%I #23 Dec 21 2016 18:45:23
%S 1,12,710,267,159,164,76,90,16285,2168,3832,7773,29849,34731,1496,
%T 23485,51130,17658,38908,38639,270845,450432,57050,145850,631632,
%U 240947,398108,306349,288481,410531,1516421,2621329,781173,333140,2997665,948049,593835,1506645,1216039
%N 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.
%C 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.
%C Conjecture: for all n>1, there exists an integer k to get 5 primes in arithmetic progression starting with k*prime(n)# - 1.
%H Pierre CAMI, <a href="/A277691/a277691.txt">PFGW Script</a>
%e 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.
%t 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 *)
%t 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 *)
%Y Cf: A002110.
%K nonn
%O 2,2
%A _Pierre CAMI_, Oct 27 2016
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