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A359272
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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).
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0
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2, 23, 2, 47, 7, 2, 251, 89, 139, 2, 1889, 199, 1627, 23, 2, 1741, 883, 18839, 47, 113, 2, 19471, 12401, 123229, 251, 5939, 89, 2, 118801, 463, 776257, 1889, 158867, 1823, 523, 2, 498259, 36551, 3873011, 1741, 894287, 20809, 15823, 139, 2, 148531, 11593, 23884639, 19471, 6996307, 73133, 74453, 1627, 1129, 2, 406951, 183091
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OFFSET
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3,1
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COMMENTS
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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.
T(m,1) = 2.
T(m,k) = T(2*m,k) if k is odd.
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LINKS
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EXAMPLE
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The array starts with row 3 as follows:
2 23 47 251 1889 1741 19471
2 7 89 199 883 12401 463
2 139 1627 18839 123229 776257 3873011
2 23 47 251 1889 1741 19471
2 113 5939 158867 894287 6996307 9984437
2 89 1823 20809 73133 989647 3250469
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).
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MAPLE
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P:= select(isprime, [2, seq(i, i=3..3*10^7, 2)]): nP:= nops(P):
M:= Matrix(20, 20):
for m from 3 to 20 do
x:= 2; a:= 1; r:= 1;
for j from 2 to nP do
v:= P[j] mod m;
if v <> a then
w:= j - r;
if M[m, w] = 0 then M[m, w]:= P[r] fi;
a:= v; r:= j;
fi;
od
od:
M[3..20, 1..10];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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