# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a083809 Showing 1-1 of 1 %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, Table of n, a(n) for n = 1..591 %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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE