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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}. 1

%I

%S 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 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 23,19,17,13,41,0,47,43,41,37,0,31,29,0,23,19,17,47

%N 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 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 2:..0...0...0

%C 3:..0...2...0...2

%C 4:..7...5...3...0...5

%C 5:..0...0...0...0...2...0

%C 6:.13..11...0...7...5...3..11

%C 7:..0...0...0...0...0...0...0...0

%C 8:.19..17...0..13..11...0...7...5..17

%C Every odd row afterward would then be entirely filled with zeros and every third column would contain zeros, often following an initial prime.

%C The triangle begins as follows:

%C b

%C --+b^2..+0..+1..+3..+4..+6..+7..+9.+10.+12

%C 2.:......0...0

%C 4.:......7...5...0...5

%C 6.:.....13..11...7...5..11

%C 8.:.....19..17..13..11...7...5

%C 10:......0..23..19..17..13..11...7..23

%C 12:.....31..29...0..23..19..17..13..11..29

%C 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 The fraction |A144912(b, b^2)| / b approaches 3 or nearly 3.

%C 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 The units digit in columns follows the repeating sequence {1, 7, 3, 9, 5}, with nearly all fives omitted and occasional other omissions.

%C The units digit in rows follows the sequence {1, 9, 5, 3, 9, 7, 3, 1, 7, 5}.

%C The complete repeating unit is:

%C 1 9 5 3 9 7 3 1 7 5

%C 7 5 1 9 5 3 9 7 3 1

%C 3 1 7 5 1 9 5 3 9 7

%C 9 7 3 1 7 5 1 9 5 3

%C 5 3 9 7 3 1 7 5 1 9

%o (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 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 Cf. A144912, A145009.

%K nonn,base,easy,tabf

%O 2,3

%A _Reikku Kulon_, Sep 25 2008

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Last modified May 28 05:47 EDT 2022. Contains 354112 sequences. (Running on oeis4.)