

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).


REFERENCES

Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, 2000.


LINKS



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}]]


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
(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(k1)): 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.


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



