A083809 Let f(n) be the smallest prime == 1 mod n (cf. A034694). Sequence gives triangle T(j,k) = f^k(j) for 1 <= k <= j, read by rows.

2, 3, 7, 7, 29, 59, 5, 11, 23, 47, 11, 23, 47, 283, 1699, 7, 29, 59, 709, 2837, 22697, 29, 59, 709, 2837, 22697, 590123, 1180247, 17, 103, 619, 2477, 34679, 416149, 7490683, 29962733, 19, 191, 383, 4597, 27583, 330997, 9267917, 74143337, 1038006719

Offset: 1

Author

Amarnath Murthy, May 08 2003

Keywords

• nonn
• tabl

Comments

It has been proved in the reference that for every prime p there exists a prime of the form k*p+1. Conjecture: sequence is infinite, i.e., for every n there exists a prime of the form n*k+1 (cf. A034693).

Both follow directly from Dirichlet's theorem. [Charles R Greathouse IV, Feb 28 2012]

Examples

			The first few rows of the triangle are
2
3 7
7 29 59
5 11 23 47
11 23 47 283 1699
7 29 59 709 2837 22697
		

References

• Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, 2000.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..591

Crossrefs

The first column is given by A034694; the sequence of the last terms in the rows (main diagonal) is A083810. Row sums are in A160940.

Cf. A034693.

Programs

Magma

f:=function(n) m:=1; while not IsPrime(m*n+1) do m+:=1; end while; return m*n+1; end function; &cat[ [ k eq 1 select f(j) else f(Self(k-1)): k in [1..j] ]: j in [1..9] ]; // Klaus Brockhaus, May 30 2009

Mathematica

f[1]=2; f[n_] := f[n] = Block[{p=2}, While[Mod[p, n] != 1, p = NextPrime[p]]; p];
Flatten[Table[Rest @ NestList[f, j, j], {j, 9}]]
(* Jean-François Alcover, May 31 2011, improved by Robert G. Wilson v *)

PARI

f(n)=my(k=n+1);while(!isprime(k),k+=n);k
T(j,k)=for(i=1,k,j=f(j));j \\ Charles R Greathouse IV, Feb 28 2012

Extensions

Edited, corrected and extended by Klaus Brockhaus, May 13 2003