A182060 Triangle read by rows arising from the Mann-Shanks criterion for primality.

1, 1, 2, 3, 4, 5, 10, 6, 6, 35, 8, 21, 56, 9, 8, 126, 120, 11, 10, 330, 792, 286, 14, 55, 792, 1287, 364, 15, 78, 2002, 6435, 4368, 680, 18, 14, 1365, 11440, 19448, 8568, 969, 20, 105, 4368, 24310, 31824, 11628, 1140, 21, 16, 2380, 31824, 92378, 77520, 20349, 1540, 23

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

Sequence in context: A092233 A115895 A116017 * A067033 A067034 A357449

Adjacent sequences: A182057 A182058 A182059 * A182061 A182062 A182063

Keyword

nonn,tabf

Author

N. J. A. Sloane, Apr 09 2012

Extensions

a(19) onwards from Andrew Howroyd, Feb 20 2024

Status

approved