A359272 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).

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

Offset

3,1

Comments

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.

Links

Table of n, a(n) for n=3..59.

Example

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).

Maple

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];

Crossrefs

Sequence in context: A374376 A329336 A323396 * A107801 A262702 A360534

Adjacent sequences: A359269 A359270 A359271 * A359273 A359274 A359275

Keyword

nonn,tabl

Author

Robert Israel, Dec 27 2022

Status

approved