cp's OEIS Frontend

A144923 Triangle read by rows: |A144912(b, b^2 + k)| if it is prime and 0 otherwise, with rows b in {2, 4, 6, ...} and columns k in {0, 1, 3, 4, 6, 7, ..., b}.

%I A144923 #11 Jul 24 2020 02:09:44
%S A144923 0,0,7,5,0,5,13,11,7,5,11,19,17,13,11,7,5,0,23,19,17,13,11,7,23,31,29,
%T A144923 0,23,19,17,13,11,29,37,0,31,29,0,23,19,17,13,11,43,41,37,0,31,29,0,
%U A144923 23,19,17,13,41,0,47,43,41,37,0,31,29,0,23,19,17,47
%N A144923 Triangle read by rows: |A144912(b, b^2 + k)| if it is prime and 0 otherwise, with rows b in {2, 4, 6, ...} and columns k in {0, 1, 3, 4, 6, 7, ..., b}.
%C A144923 This triangle is roughly twice the usual width. Odd rows and columns congruent to 2 modulo 3 are omitted; otherwise the triangle would begin like this:
%C A144923 2:..0...0...0
%C A144923 3:..0...2...0...2
%C A144923 4:..7...5...3...0...5
%C A144923 5:..0...0...0...0...2...0
%C A144923 6:.13..11...0...7...5...3..11
%C A144923 7:..0...0...0...0...0...0...0...0
%C A144923 8:.19..17...0..13..11...0...7...5..17
%C A144923 Every odd row afterward would then be entirely filled with zeros and every third column would contain zeros, often following an initial prime.
%C A144923 The triangle begins as follows:
%C A144923 b
%C A144923 --+b^2..+0..+1..+3..+4..+6..+7..+9.+10.+12
%C A144923 2.:......0...0
%C A144923 4.:......7...5...0...5
%C A144923 6.:.....13..11...7...5..11
%C A144923 8.:.....19..17..13..11...7...5
%C A144923 10:......0..23..19..17..13..11...7..23
%C A144923 12:.....31..29...0..23..19..17..13..11..29
%C A144923 Some diagonals are entirely filled with zeros; for example, the first such diagonal begins at b = 32 and there is another for b in [40, 42].
%C A144923 The fraction |A144912(b, b^2)| / b approaches 3 or nearly 3.
%C A144923 For n = b and m = b + 2, ((n, x) + (m, x)) / 2 approximates (m, x + 1) = (n, x - 1), where x is the index of a column disregarding k.
%C A144923 The units digit in columns follows the repeating sequence {1, 7, 3, 9, 5}, with nearly all fives omitted and occasional other omissions.
%C A144923 The units digit in rows follows the sequence {1, 9, 5, 3, 9, 7, 3, 1, 7, 5}.
%C A144923 The complete repeating unit is:
%C A144923 1 9 5 3 9 7 3 1 7 5
%C A144923 7 5 1 9 5 3 9 7 3 1
%C A144923 3 1 7 5 1 9 5 3 9 7
%C A144923 9 7 3 1 7 5 1 9 5 3
%C A144923 5 3 9 7 3 1 7 5 1 9
%o A144923 (PARI) T(b, k) = {my(d=digits(k, b)); if(isprime(d=abs(sum(i=1, #d, 2*d[i]-b+1))), d, 0); }
%o A144923 row(n) = {my(v=[]); for(k=0, 2*n, if(k%3<2, v=concat(v, T(2*n, 4*n^2+k)))); v; } \\ _Jinyuan Wang_, Jul 21 2020
%Y A144923 Cf. A144912, A145009.
%K A144923 nonn,base,easy,tabf
%O A144923 2,3
%A A144923 _Reikku Kulon_, Sep 25 2008