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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.

%I A083809 #21 Sep 08 2022 08:45:10
%S A083809 2,3,7,7,29,59,5,11,23,47,11,23,47,283,1699,7,29,59,709,2837,22697,29,
%T A083809 59,709,2837,22697,590123,1180247,17,103,619,2477,34679,416149,
%U A083809 7490683,29962733,19,191,383,4597,27583,330997,9267917,74143337,1038006719
%N 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.
%C A083809 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).
%C A083809 Both follow directly from Dirichlet's theorem. [_Charles R Greathouse IV_, Feb 28 2012]
%D A083809 Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, 2000.
%H A083809 Vincenzo Librandi, <a href="/A083809/b083809.txt">Table of n, a(n) for n = 1..591</a>
%e A083809 The first few rows of the triangle are
%e A083809 2
%e A083809 3 7
%e A083809 7 29 59
%e A083809 5 11 23 47
%e A083809 11 23 47 283 1699
%e A083809 7 29 59 709 2837 22697
%t A083809 f[1]=2; f[n_] := f[n] = Block[{p=2}, While[Mod[p, n] != 1, p = NextPrime[p]]; p];
%t A083809 Flatten[Table[Rest @ NestList[f, j, j], {j, 9}]]
%t A083809 (* _Jean-François Alcover_, May 31 2011, improved by _Robert G. Wilson v_ *)
%o A083809 (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,",")))
%o A083809 (PARI) f(n)=my(k=n+1);while(!isprime(k),k+=n);k
%o A083809 T(j,k)=for(i=1,k,j=f(j));j \\ _Charles R Greathouse IV_, Feb 28 2012
%o A083809 (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
%Y A083809 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.
%Y A083809 Cf. A034693.
%K A083809 nonn,tabl
%O A083809 1,1
%A A083809 _Amarnath Murthy_, May 08 2003
%E A083809 Edited, corrected and extended by _Klaus Brockhaus_, May 13 2003