

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



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, 0, 23, 19, 17, 13, 11, 29, 37, 0, 31, 29, 0, 23, 19, 17, 13, 11, 43, 41, 37, 0, 31, 29, 0, 23, 19, 17, 13, 41, 0, 47, 43, 41, 37, 0, 31, 29, 0, 23, 19, 17, 47
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

2,3


COMMENTS

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:
2:..0...0...0
3:..0...2...0...2
4:..7...5...3...0...5
5:..0...0...0...0...2...0
6:.13..11...0...7...5...3..11
7:..0...0...0...0...0...0...0...0
8:.19..17...0..13..11...0...7...5..17
Every odd row afterward would then be entirely filled with zeros and every third column would contain zeros, often following an initial prime.
The triangle begins as follows:
b
+b^2..+0..+1..+3..+4..+6..+7..+9.+10.+12
2.:......0...0
4.:......7...5...0...5
6.:.....13..11...7...5..11
8.:.....19..17..13..11...7...5
10:......0..23..19..17..13..11...7..23
12:.....31..29...0..23..19..17..13..11..29
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].
The fraction A144912(b, b^2) / b approaches 3 or nearly 3.
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.
The units digit in columns follows the repeating sequence {1, 7, 3, 9, 5}, with nearly all fives omitted and occasional other omissions.
The units digit in rows follows the sequence {1, 9, 5, 3, 9, 7, 3, 1, 7, 5}.
The complete repeating unit is:
1 9 5 3 9 7 3 1 7 5
7 5 1 9 5 3 9 7 3 1
3 1 7 5 1 9 5 3 9 7
9 7 3 1 7 5 1 9 5 3
5 3 9 7 3 1 7 5 1 9


LINKS

Table of n, a(n) for n=2..70.


PROG

(PARI) T(b, k) = {my(d=digits(k, b)); if(isprime(d=abs(sum(i=1, #d, 2*d[i]b+1))), d, 0); }
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


CROSSREFS

Cf. A144912, A145009.
Sequence in context: A241902 A113223 A096414 * A184908 A197519 A290374
Adjacent sequences: A144920 A144921 A144922 * A144924 A144925 A144926


KEYWORD

nonn,base,easy,tabf


AUTHOR

Reikku Kulon, Sep 25 2008


STATUS

approved



