1,1

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?

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;

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

Sequence in context: A018313 A074820 A006169 this_sequence A102643 A125631 A045408

Adjacent sequences: A088325 A088326 A088327 this_sequence A088329 A088330 A088331

nonn,tabf

Pierre CAMI (colettecami(AT)aol.com), Nov 06 2003

Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 30 2009