OEIS A401136

The sequence

OEIS A401136 (proposed) is the minus face of the grid, read as one sequence: the square array

$Y(n,k)=\mathrm{A}002110(n-1)\cdot \mathrm{prime}(n{)}^k-1,n,k\ge 1,$

read by antidiagonals downwards. In an OEIS entry $n$ is always the index, so the entry writes the order as $n$ and the exponent as $k$. This site and the paper use the opposite letters: the entry's $Y(n,k)$ is $Y_n(k)=p_{n-1}\mathrm{\#}\cdot p_n^k-1$.

The array begins:

   1,     3,      7,      15,       31, ...
   5,    17,     53,     161,      485, ...
  29,   149,    749,    3749,    18749, ...
 209,  1469,  10289,   72029,   504209, ...
2309, 25409, 279509, 3074609, 33820709, ...
...

Row 1 consists of the Mersenne numbers $2^k-1$ (A000225), and column 1 is the primorial numbers minus one, $\mathrm{A}002110(n)-1$ (A057588). Rows 2 and 3 are A048473 and A198764 from their second terms on (both entries start at index 0). For example, for $n=4$ the product of the first three primes is 30 and $\mathrm{prime}(4)=7$, so $Y(4,2)=30\cdot 7^2-1=1469$.

Antidiagonal $d$ holds $Y(1,d),Y(2,d-1),\dots ,Y(d,1)$: it opens with the Mersenne number $2^d-1$ and closes with $\mathrm{A}002110(d)-1$. The cell $Y(n,k)$ is term number $(n+k-1)(n+k-2)/2+n$, and the sequence begins 1; 3, 5; 7, 17, 29; 15, 53, 149, 209; … Along each row, $Y(n,k)=\mathrm{prime}(n)\cdot Y(n,k-1)+\mathrm{prime}(n)-1$ for $k\ge 2$.

A400659$(n)$ is the position of the first prime in row $n$ of this array. Which cells are prime is the subject of the catalogue of proven primes; any single cell can be computed exactly on the datasets page, for example $Y_4(2)=1469$.

The b-file

The b-file attached to the entry lists 9,870 terms: the first 140 antidiagonals, complete. The largest has 333 digits. Every term is an exact integer computed from the definition; no primality statement is involved.

Check it yourself

Two self-contained programs recompute every term of the b-file on your own computer. Neither needs any data from this site:

• A401136.gp, for PARI/GP;
• A401136.py, for Python.

Each computes the first 140 antidiagonals from the definition and checks them against the 36 terms of the entry's data, row 1 against $2^k-1$, column 1 against the product of the first $n$ primes minus one, and the row recurrence at every cell with $k\ge 2$. The Python program also compares a SHA-256 fingerprint of all 9,870 terms with the b-file uploaded to the OEIS, so it checks the whole file without downloading it. If the b-file is in the current directory (as b401136.txt or b401136_1.txt), both programs also compare every term with it, line by line.

gp -q A401136.gp
python A401136.py

The Python program prints

A401136: 9870 terms = antidiagonals 1..140, largest 333 digits
DATA (36 terms): ok
row 1 = 2^k - 1 for k = 1..140: ok
column 1 = A002110(n) - 1 for n = 1..140: ok
Y(n,k) = prime(n)*Y(n,k-1) + prime(n) - 1 at all 9730 cells with k >= 2: ok
SHA-256 of the 9870 lines = the b-file uploaded to the OEIS: ok
b-file: none in this directory (optional; the fingerprint above covers it)
OK: all checks passed. Total 0 s.

The gp program makes the same checks except the fingerprint. Each run takes well under a second.

Requirements. PARI/GP (tested with version 2.17.3), or Python 3.8 or later with the standard library only. Both are free software.