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%I #33 Dec 31 2022 02:32:12
%S 2,23,2,47,7,2,251,89,139,2,1889,199,1627,23,2,1741,883,18839,47,113,
%T 2,19471,12401,123229,251,5939,89,2,118801,463,776257,1889,158867,
%U 1823,523,2,498259,36551,3873011,1741,894287,20809,15823,139,2,148531,11593,23884639,19471,6996307,73133,74453,1627,1129,2,406951,183091
%N Array read by downward antidiagonals: for m >= 3 and n >= 1, T(m,n) is the first prime that starts a string of exactly n consecutive primes that are congruent (mod m).
%C T(m,n) is the first prime prime(k) such that prime(k) == prime(j) (mod m) for k <= j <= k+n-1 but not for j = k-1 or k+n.
%C T(m,1) = 2.
%C T(m,k) = T(2*m,k) if k is odd.
%e The array starts with row 3 as follows:
%e 2 23 47 251 1889 1741 19471
%e 2 7 89 199 883 12401 463
%e 2 139 1627 18839 123229 776257 3873011
%e 2 23 47 251 1889 1741 19471
%e 2 113 5939 158867 894287 6996307 9984437
%e 2 89 1823 20809 73133 989647 3250469
%e T(3,4) = 251 because the 4 consecutive primes starting at 251 are 251,257,263,269, which are all congruent mod 3, but the previous prime 241 and the next prime 271 are not congruent to 251 (mod 3).
%p P:= select(isprime,[2,seq(i,i=3..3*10^7,2)]): nP:= nops(P):
%p M:= Matrix(20,20):
%p for m from 3 to 20 do
%p x:= 2; a:= 1; r:= 1;
%p for j from 2 to nP do
%p v:= P[j] mod m;
%p if v <> a then
%p w:= j - r;
%p if M[m,w] = 0 then M[m,w]:= P[r] fi;
%p a:= v; r:= j;
%p fi;
%p od
%p od:
%p M[3..20,1..10];
%K nonn,tabl
%O 3,1
%A _Robert Israel_, Dec 27 2022
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