OFFSET
1,1
COMMENTS
A k X k 'Futility Square' is a stack of k primes, each one being of length k. The 1st horizontal prime is also the 1st vertical one; the 2nd horizontal prime is also the 2nd vertical one, and so on. The first horizontal prime must be zeroless. The number on the main diagonal (running from top left to bottom right) is also a prime. The k + 1 primes involved in a k X k square must be distinct and the smallest possible not leading to a contradiction.
There might be more than one k X k 'Futility Square' for some k >= 2. For example another such square for k = 2 is
.
6 7
7 1
. - David A. Corneth, Dec 23 2023
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..5049 (through k=101; terms 1..44 from Michael S. Branicky, terms 45..1539 from David A. Corneth)
Éric Angelini, Squaring Primes, Personal blog, December 2023.
Michael S. Branicky, Python program
David A. Corneth, PARI program
Futility Closet, Squaring Words, Futility Closet, December 2023.
EXAMPLE
Here is the lexicographically earliest 3 X 3 'Futility Square':
.
1 1 3
1 0 1
3 1 3
.
We see that 113, 101, 313 and the diagonal 103 are distinct primes.
Hereunder is the lexicographically earliest 6 X 6 'Futility Square':
.
1 1 1 1 1 9
1 0 0 0 0 3
1 0 0 0 1 9
1 0 0 1 2 9
1 0 1 2 0 7
9 3 9 9 7 3
.
We see that 111119, 100003, 100019, 100129, 101207, 939973 and the diagonal 100103 are distinct primes.
The sequence is formed by the two horizontal primes of the 2 X 2 square [13, 31], then the three horizontal primes of the 3 X 3 square [113, 101, 313], then the four horizontal primes of the 4 X 4 square [1117, 1009, 1019, 7993], etc.
PROG
(PARI) \\ See PARI link
(Python) # See Python link
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Eric Angelini and Giorgos Kalogeropoulos, Dec 09 2023
EXTENSIONS
a(28)-a(35) for k=8 from Michael S. Branicky, Dec 23 2023
STATUS
approved