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Triangle read by rows: T(n,k) is the least prime == k (mod prime(n)), for 0 <= k < prime(n).
2

%I #11 Jan 20 2021 18:45:03

%S 2,3,3,7,2,5,11,2,3,19,7,29,2,3,11,5,13,11,23,2,3,37,5,17,7,19,31,43,

%T 13,53,2,3,17,5,19,7,47,61,23,11,103,17,103,2,3,89,5,23,7,59,43,61,11,

%U 29,13,31,83,67,19,191,2,3,23,5,101,7,103,47,29,11,31,13,71,53,73,17,37

%N Triangle read by rows: T(n,k) is the least prime == k (mod prime(n)), for 0 <= k < prime(n).

%C By Dirichlet's theorem on primes in arithmetic progressions, T(n,k) always exists.

%C T(n,0) = prime(n).

%H Robert Israel, <a href="/A340753/b340753.txt">Table of n, a(n) for n = 1..10191</a> (rows 1 to 68, flattened)

%e T(4,6) = 13 because prime(4) = 7 and 13 is the first prime == 6 (mod 7).

%e Triangle begins:

%e 2, 3

%e 3, 7, 2

%e 5, 11, 2, 3, 19

%e 7, 29, 2, 3, 11, 5, 13

%e 11, 23, 2, 3, 37, 5, 17, 7, 19, 31, 43

%e 13, 53, 2, 3, 17, 5, 19, 7, 47, 61, 23, 11, 103

%e 17, 103, 2, 3, 89, 5, 23, 7, 59, 43, 61, 11, 29, 13, 31, 83, 67

%p h:= proc(p) local A, count,q,v;

%p A:= Array(0..p-1);

%p q:= 1; count:= 0;

%p while count < p do

%p q:= nextprime(q);

%p v:= q mod p;

%p if A[v] = 0 then count:= count+1; A[v]:= q; fi

%p od;

%p op(convert(A,list));

%p end proc:

%p for i from 1 to 20 do h(ithprime(i)) od;

%Y Cf. A340752.

%K nonn,tabf,look

%O 1,1

%A _Robert Israel_, Jan 19 2021