Woodall Numbers

Today, having turned 26906 days old, I discovered that this number has a connection with so-called Woodall primes that are simply Woodall numbers that are prime. So what characterises a Woodall number? To quote from this source:

In number theory, a Woodall number $W_n$ is any natural number of the form:$$W_n=n\cdot 2^n-1$$for some natural number $n$. The first few Woodall numbers are: 1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in the OEIS). Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly defined Cullen numbers. The previous source goes on to say that:

Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents $n$ for which the corresponding Woodall numbers $W_n$ are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... (sequence A002234 in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... (sequence A050918 in the OEIS) ... almost all Woodall numbers are composite. It is an open problem whether there are infinitely many Woodall primes. As of October 2018, the largest known Woodall prime is $17016602\times 2^{17016602}-\hspace{0.17em}1$. It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.

26906 however, to return to my diurnal age, arises in the context of so-called Generalized Woodall Numbers. These numbers are defined thus:

A generalized Woodall number base b is defined to be a number of the form $n\times b^n-1$, where $n+2>b$ if a prime can be written in this form, it is then called a generalized Woodall prime.

The number associated with my diurnal age turns out to be a member of OEIS A299378:

A299378

Numbers $k$ such that $k\cdot {15}^k-1$ is prime.

Thus it can be seen that 26906 belongs to those exponents for which the generalized Woodall numbers $k\cdot {15}^k-1$, with $b=15$,  are prime. These are quite rare with the initial exponents being:

2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606

Thus $26906\cdot {15}^{26906}-1$ is prime and is needless to say a very large prime. As of November 2021, the largest known generalized Woodall prime with base greater than 2 is $2740879\cdot {32}^{2740879}-1$.

Below is a list of numbers $n$ for increasing values of $b$ such that $n\cdot b^n-1$ is prime. The corresponding OEIS sequence is listed if it exists:

1 3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284, 294, ... (all primes plus 1) A008864

2 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018, 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, ... A002234

3 1, 2, 6, 10, 18, 40, 46, 86, 118, 170, 1172, 1698, 1810, 2268, 4338, 18362, 72662, 88392, 94110, 161538, 168660, 292340, 401208, 560750, 1035092, ... A006553

4 1, 2, 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, 293912, ..., 1993191, ... A086661

5 8, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, 349656, 545082, ... A059676

6 1, 2, 3, 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, 145359, 249987, ... A059675

7 2, 18, 68, 84, 3812, 14838, 51582, ... A242200

8 1, 2, 7, 12, 25, 44, 219, 252, 507, 1155, 2259, 2972, 4584, 12422, 13905, 75606, ... A242201

9 10, 58, 264, 1568, 4198, 24500, ... A242202

10 2, 3, 8, 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, 4665, 5364, ... A059671

11 2, 8, 252, 1184, 1308, ... A299374

12 1, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, ... A299375

13 2, 6, 563528, ... A299376

14 1, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, ... A299377

15 2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, ... A299378

16 167, 189, 639, ... A299379

17 2, 18, 20, 38, 68, 3122, 3488, 39500, ... A299380

18 1, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ... A299381

19 12, 410, 33890, 91850, 146478, 189620, 280524, ... A299382

20 1, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129, ... A299383

21 2, 18, 200, 282, 294, 1174, 2492, 4348, ...

22 2, 5, 140, 158, 263, 795, 992, 341351, ...

23 29028, ...

24 1, 2, 5, 12, 124, 1483, 22075, 29673, 64593, ...

25 2, 68, 104, 450, ...

26 3, 8, 79, 132, 243, 373, 720, 1818, 11904, 134778, ...

27 10, 18, 20, 2420, 6638, 11368, 14040, 103444, ...

28 2, 5, 6, 12, 20, 47, 71, 624, 1149, 2399, 8048, 30650, 39161, ...

29 26850, 237438, 272970, ...

30 1, 63, 331, 366, 1461, 3493, 4002, 5940, 13572, 34992, 182461, 201038, ...

Let's take some examples for $b=30$ from the above list. When $n=1$, we have $1\cdot {30}^1-1=29$ which is prime and when $n=63$ we have $63\cdot {30}^{63}-1$ being prime. This latter number is:

72107360226142762177814830874900999999999999999999999999999999999999999999999999999999999999999