OFFSET
1,3
COMMENTS
Take Pascal's triangle (A007318) with each row shifted right by two more places (see Examples section). Color an entry red if it is divisible by the row number. Mann and Shanks prove that the number of a column is prime iff every entry in that column is red.
The sequence is the triangle obtained by reading down the prime-numbered columns. For example, column 13 is 10 followed by 6, both numbers being red. See link for a better picture.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..4024 (rows 1..100)
Henry B. Mann and Daniel Shanks, A necessary and sufficient condition for primality, and its source, J. Combinatorial Theory Ser. A13 (1972), 131--134. MR0306098 (46 #5225)
N. J. A. Sloane, Illustration of initial terms
EXAMPLE
Pascal's triangle with each row shifted right by two more places:
1
., ., 1, 1
., ., ., ., 1, 2, 1
., ., ., ., ., ., 1, 3, 3, 1
., ., ., ., ., ., ., ., 1, 4, 6, 4, 1
., ., ., ., ., ., ., ., ., ., 1, 5,10,10, 5, 1
., ., ., ., ., ., ., ., ., ., ., ., 1, 6,15,20,15, 6, 1
., ., ., ., ., ., ., ., ., ., ., ., ., ., 1, 7,21,35,35,21, 7, 1
., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., 1, 8,28,56,70,56, 28, 8,...
., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., 1,9,36,84,126,126,...
., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., 1,10, 45,120,...
., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., .1,.11,...
Columns (written horizontally) with prime column numbers:
1
1
2
3
4 5
10 6
6 35 8
21 56 9
8 126 120 11
...
See the link for a better illustration.
PROG
(PARI) \\ B(k) gives k-th column in relevant range.
B(k)={my(b=(k-1)\3); vector(k\2-b, j, binomial(b+j, k-2*(b+j)))}
T(n)=[B(p) | p<-primes(n)]
{my(A=T(10)); for(i=1, #A, print(A[i]))} \\ Andrew Howroyd, Feb 20 2024
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Apr 09 2012
EXTENSIONS
a(19) onwards from Andrew Howroyd, Feb 20 2024
STATUS
approved