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A083809
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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.
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3
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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; internal format)
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OFFSET
| 1,1
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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).
The first column is given by A034694; the sequence of the last terms in the rows (main diagonal) is A083810.
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REFERENCES
| Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11,2000.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..591
M. L. Perez et al., eds., Smarandache Notions Journal
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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
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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}]]
(* From J.F.Alcover, May 31 2011, improved by Robert G. Wilson *)
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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, ", ")))
(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] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), May 30 2009]
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CROSSREFS
| Cf. A034693, A034694, A083810.
Row sums are in A160940. [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), May 30 2009]
Sequence in context: A179894 A027672 A104138 * A092967 A056431 A199466
Adjacent sequences: A083806 A083807 A083808 * A083810 A083811 A083812
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KEYWORD
| nonn,tabl
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 08 2003
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EXTENSIONS
| Edited, corrected and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), May 13 2003
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