A368051 Successive primes building the lexicographically earliest k X k 'Futility Squares' (see the Comment and Example sections for more explanations).

13, 31, 113, 101, 313, 1117, 1009, 1019, 7993, 11113, 10007, 10009, 10039, 37997, 111119, 100003, 100019, 100129, 101207, 939973, 1111151, 1000003, 1000033, 1000037, 1000099, 5033981, 1337911, 11111117, 10000019, 10000079, 10000103, 10000121, 10011163, 11702641, 79931311

Offset: 1

Author

Eric Angelini and Giorgos Kalogeropoulos, Dec 09 2023

Keywords

• base
• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A038618, A091633.

Programs

PARI

\\ See PARI link

Python

# See Python link

Extensions

a(28)-a(35) for k=8 from Michael S. Branicky, Dec 23 2023