Woodall numbers

A number of the form $W(n)=n\cdot2^n-1$ is called Woodall number.

Woodall numbers are also called Riesel numbers.

Some infinite sums

$\sum_{n=1}^{\infty}\frac{1}{W(n)+1}=\log 2,\quad\quad% \sum_{n=0}^{\infty}\frac{W(n)}{n!}=2e^2-e\,,$

and

$\sum_{n=1}^{\infty}% W(n){{2n}\choose n}^{-1}=\frac{8}{3}+\pi+\frac{2\pi}{9\sqrt{3}}.$

A number of the form $W_b(n)=n\cdot b^n-1$ , for $n \ge b-1$ , is called generalized Woodall number.

If the bound $n \ge b-1$ is disregarded, then we have the following nice double infinite sum

$\sum_{n=2}^{\infty}\sum_{b=2}^{\infty}\frac{1}{1+W_b(n)} = 1-\gamma\,,$

where $\gamma$ is the Euler-Mascheroni constant.

The first Woodall numbers are 1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575

The first generalized Woodall numbers are 1, 7, 17, 23, 63, 80, 159, 191, 323, 383, 895, 1023, 1214, 2047, 2499, 4373, 4607, 5119, 10239, 15308, 15624, 22527, 24575, 38879, 49151, 52487, 93749 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

remainders of Generalized Woodall numbers

A graph displaying how many Generalized Woodall numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

divisibility of Generalized Woodall numbers

Useful links Mathworld, Woodall Number
Wikipedia, Woodall number
OEIS, Sequence A003261

Generalized Woodall numbers can also be... (you may click on names or numbers and on + to get more values)

aban 17 23 63 80 159 191 323 + 895 abundant 80 15624 5764800 alternating 23 63 323 383 1214 amenable 17 80 4373 15308 15624 93749 590489 + 215233604 apocalyptic 2047 4607 5119 10239 15308 15624 22527 24575 arithmetic 17 23 159 191 323 383 895 + 9961471 brilliant 323 2047 93749 90699263 Chen 17 23 191 524287 590489 congruent 23 63 80 159 191 323 383 + 9961471 constructible 17 80 Cunningham 17 63 80 323 1023 2047 2499 + 531726889113615 Curzon 4373 590489 cyclic 17 23 159 191 323 383 895 + 9961471 D-number 63 159 1023 10239 d-powerful 63 4373 2359295 de Polignac 38879 49151 52487 590489 14680063 46137343 decagonal 2047 deficient 17 23 63 159 191 323 383 + 9961471 dig.balanced 2499 52487 705893 10485759 Duffinian 63 323 2047 22527 24575 38879 49151 + 9961471 economical 17 23 159 191 383 1214 2047 + 14680063 emirp 17 emirpimes 159 1214 2047 93749 10485759 90699263 equidigital 17 23 159 191 383 1214 2047 + 14680063 esthetic 23 323 evil 17 23 63 80 159 323 383 + 872415231 fibodiv 323 Friedman 15624 114687 177146 279935 524287 frugal 229375 gapful 15624 24575 229375 491519 1048575 1959551 5764800 + 68719476735 good prime 17 191 happy 23 383 15624 22527 93749 114687 1959551 5764800 Harshad 63 80 4607 15624 5764800 3486784400 hoax 895 hungry 17 iban 17 23 323 1023 1214 2047 4373 inconsummate 63 383 4373 15308 114687 279935 491519 interprime 15624 1948616 10485759 junction 15624 279935 491519 97656249 Kaprekar 99999999999 katadrome 63 80 Kynea 23 Lehmer 2047 Leyland 17 lonely 23 lucky 63 159 895 1023 10239 22527 524287 m-pointer 23 magnanimous 23 metadrome 17 23 159 modest 23 2499 Moran 63 4607 Motzkin 323 nialpdrome 63 80 99999999999 oban 17 23 63 80 323 383 895 odious 191 895 1214 2047 4373 5119 15308 + 838860799 palindromic 191 323 383 99999999999 palprime 191 383 pancake 191 panconsummate 23 pernicious 17 80 191 1214 2047 4373 5119 + 5764800 Perrin 17 Pierpont 17 plaindrome 17 23 159 2499 8999999999 99999999999 11999999999999 129999999999999 Poulet 2047 137438953471 practical 80 15624 5764800 prime 17 23 191 383 4373 5119 524287 + 824633720831 Proth 17 pseudoperfect 80 15624 repdigit 99999999999 repunit 63 1023 2047 524287 1048575 134217727 68719476735 137438953471 self 323 24575 114687 2228223 9961471 semiprime 159 323 895 1214 2047 4607 10239 + 90699263 Smith 895 Sophie Germain 23 191 4373 590489 sphenic 1023 177146 546874 705893 2228223 9961471 46118407 + 66961565 straight-line 159 99999999999 strong prime 17 191 590489 14680063 super Niven 80 super-d 5119 106495 229375 491519 9961471 tau 80 1948616 trimorphic 99999999999 truncatable prime 17 23 383 4373 twin 17 191 3124999 uban 17 23 63 80 Ulam 1023 49151 93749 491519 3124999 9961471 undulating 191 323 383 unprimeable 895 106495 177146 279935 546874 2359295 5764800 untouchable 15624 upside-down 159 wasteful 63 80 323 895 1023 2499 4607 + 9961471 weak prime 23 383 4373 5119 524287 3124999 Zumkeller 80 15624 zygodrome 99999999999 11999999999999