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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. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 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=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

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Last modified September 27 08:15 EDT 2023. Contains 365674 sequences. (Running on oeis4.)