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

1,1

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]

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

Example

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

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 *)

Prog

(PARI 2.1.3) for(j=1, 9, q=j; for(k=1, j, m=1; while(!isprime(p=m*q+1, 1), m++); print1(q=p, ", ")))
(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
(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

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.

Sequence in context: A322138 A104138 A281909 * A092967 A056431 A281947

Adjacent sequences: A083806 A083807 A083808 * A083810 A083811 A083812

Keyword

nonn,tabl

Author

Amarnath Murthy, May 08 2003

Extensions

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

Status

approved