A088328 Table read by rows where row n contains lower twin primes of the form k*A002110(n)-1 in the range 0 < k < A006094(n+1).

3, 5, 11, 17, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 29, 59, 149, 179, 239, 269, 419, 569, 599, 659, 809, 1019, 1049, 1229, 1319, 1619, 1949, 2129, 419, 1049, 2309, 2729, 3359, 5879, 6089, 6299, 7349, 7559, 8819, 9239, 10499, 10709

Offset: 1

Author

Pierre CAMI, Nov 06 2003

Keywords

• nonn
• tabf

Comments

Some of the values k*primorial(n)-1 generated by k in the range 1 to prime(n+1)*prime(n+2)-1 are not lower twin primes, A001359, so the list of k that produces the n-th row of the irregular table, as shown in A088329, is not a list of necessarily consecutive integers.

If n>2 the count of k values is near or greater than 4*log(4*p(n+1)); is this related to a proof of the infinity of twin prime pairs?

Examples

			2*2 -1 = 3, k=2, n=1
3*2 -1 = 5, k=3, n=1
6*2 -1 = 11, k=6, n=1
9*2 -1 = 17, k=9, n=1
The first three rows are:
3,5,11,17; generated by k=2, 3, 6, 9
5,11,17,29,41,59,71,101,107,137,149,179,191,197;
29,59,149,179,239,269,419,569,599,659,809,1019,1049,1229,1289,1319,1619,1949,2129;
		

Programs

Maple

isA001359 := proc(n) option remember ; return isprime(n) and isprime(n+2) ; end:
A002110 := proc(n) local i ; if n = 0 then 1; else mul(ithprime(i),i=1..n) ; end if; end proc:
A006094 := proc(n) return ithprime(n)*ithprime(n+1) ; end proc:
A088328 := proc(n,k) option remember; for j from 1 to A006094(n+1)-1 do a := j*A002110(n)-1 ; if isA001359(a) and k =1 then return a ; elif isA001359(a) and a > procname(n,k-1) then return a ; end if; end do; return -1 ; end proc:
for n from 1 to 10 do for k from 1 do T := A088328(n,k) ; if T < 0 then break; else printf("%d,",T) ; end if; end do; printf("\n") ; od: # R. J. Mathar, Oct 30 2009

Extensions

Edited by R. J. Mathar, Oct 30 2009