A002234 -id:A002234 - cp's OEIS frontend

A366898 Number of divisors of n*2^n - 1, the Woodall (or Riesel) numbers.

1, 2, 2, 6, 4, 2, 4, 4, 4, 4, 6, 4, 12, 10, 8, 48, 8, 4, 8, 4, 16, 16, 8, 8, 4, 8, 8, 16, 24, 2, 8, 4, 8, 32, 8, 32, 4, 8, 4, 24, 16, 8, 32, 8, 16, 24, 40, 16, 16, 8, 8, 16, 24, 8, 16, 8, 16, 6, 32, 8, 8, 16, 8, 512, 48, 16, 12, 48, 16, 8, 8, 4, 24, 16, 2, 256

Offset: 1

Tyler Busby, Oct 26 2023

nonn

Amiram Eldar, Table of n, a(n) for n = 1..865

Cf. A000005, A003261, A002234, A242273, A063515, A056821, A367006, A366899.

Table[DivisorSigma[0, n*2^n - 1], {n, 1, 100}] (* Amiram Eldar, Dec 11 2023 *)

a(n) = numdiv(n*2^n - 1); \\ Amiram Eldar, Dec 11 2023

a(n) = sigma0(n*2^n - 1) = A000005(A003261(n)).

A382811 Integers k such that d*2^k - 1 is prime for some divisor d of k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 12, 13, 16, 17, 18, 19, 21, 28, 30, 31, 36, 42, 46, 54, 60, 61, 63, 75, 81, 88, 89, 99, 102, 104, 106, 107, 108, 115, 123, 126, 127, 132, 133, 204, 214, 216, 225, 249, 264, 270, 286, 304, 306, 324, 330, 342, 352, 362, 384, 390

Offset: 1

Juri-Stepan Gerasimov, Apr 15 2025

nonn

			4 is in the sequence because 2*2^4 - 1 = 31 is prime for divisor d = 2 of k = 4.

Robert Israel, Table of n, a(n) for n = 1..161

Supersequence of A000043, A002234.

[k: k in [1..400] | not #[d: d in Divisors(k) | IsPrime(d*2^k-1)] eq 0];

filter:= proc(k) ormap(d -> isprime(d*2^k-1),numtheory:-divisors(k)) end proc:
select(filter, [$1..700]); # Robert Israel, Apr 25 2025

q[k_] := AnyTrue[Divisors[k], PrimeQ[#*2^k - 1] &]; Select[Range[400], q] (* Amiram Eldar, Apr 16 2025 *)

isok(k) = fordiv(k, d, if (ispseudoprime(d*2^k-1), return(1))); \\ Michel Marcus, Apr 16 2025

A367464 Numbers k such that k^5*2^k - 1 is a prime.

Original entry on oeis.org

2, 6, 9, 18, 42, 132, 139, 482, 523, 524, 859, 909, 948, 979, 1158, 1741, 2364, 3519, 4388, 5952, 6266, 8564, 12169, 14448, 54944, 103526, 116563, 125918

Offset: 1

Juri-Stepan Gerasimov, Nov 18 2023

Numbers k such that k^m*2^k - 1 is a prime: A000043 (m = 0), A002234 (m = 1), A058781 (m = 2), A367037 (m = 3), A367102 (m = 4), this sequence (m = 5).

Cf. A367421.

[k: k in [1..1000] | IsPrime(k^5*2^k-1)];

Select[Range[2500], PrimeQ[#^5*2^# - 1] &] (* Amiram Eldar, Nov 19 2023 *)

a(24)-a(25) from Michael S. Branicky, Nov 18 2023

a(26)-a(28) from Michael S. Branicky, Aug 26 2024

A367478 Numbers k such that k^6*2^k - 1 is a prime.

Original entry on oeis.org

37, 43, 167, 217, 239, 349, 581, 1297, 5183, 9119, 10589, 15205, 18745, 25687, 26609, 33667, 35663, 73603, 82501, 89269

Offset: 1

Juri-Stepan Gerasimov, Nov 19 2023

Numbers k such that k^m*2^k - 1 is a prime: A000043 (m = 0), A002234 (m = 1), A058781 (m = 2), A367037 (m = 3), A367102 (m = 4), A367464 (m = 5), this sequence (m = 6).

[k: k in [1..1000] | IsPrime(k^6*2^k-1)];

Select[Range[6000], PrimeQ[#^6*2^# - 1] &] (* Amiram Eldar, Nov 19 2023 *)

a(12) inserted by and a(14)-a(17) from Michael S. Branicky, Nov 19 2023

a(18)-a(20) from Michael S. Branicky, Nov 21 2023

A210340 Generalized Woodall primes: any primes that can be written in the form n*b^n - 1 with n+2 > b > 2.

Original entry on oeis.org

17, 191, 4373, 5119, 524287, 590489, 3124999, 14680063, 3758096383, 6973568801, 34867844009, 85449218749, 824633720831, 1099999999999, 1618481116086271, 11577835060199423, 14999999999999999, 29311444762388081, 73123168801259519

Offset: 1

Arkadiusz Wesolowski, Mar 20 2012

nonn

			167*2^668 - 1 is a prime number and 167*2^668 - 1 = 167*16^167 - 1, so this number is in the sequence.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..170
Chris Caldwell, The Top 20 Generalized Woodall Primes
Chris Caldwell, The Prime Glossary, Woodall prime
G. L. Honaker, Jr. and Chris Caldwell, 17962...40287 (1006-digits)
G. L. Honaker, Jr. and Chris Caldwell, 19981...99999 (10028-digits)
PrimeGrid, Home Page
Wikipedia, Woodall number

Cf. A050918, A002234, A006553, A086661, A059676, A059675.

lst = {}; Do[p = n*b^n - 1; If[p < 10^200 && PrimeQ[p], AppendTo[lst, p]], {b, 3, 100}, {n, b - 1, 413}]; Sort@lst

A242115 Woodall semiprimes: Semiprimes of the form n*2^n - 1.

Original entry on oeis.org

159, 895, 2047, 4607, 10239, 49151, 4718591, 20971519, 838860799, 137438953471, 5085241278463, 21440476741631, 340010386766614455386111, 96714065569170333976494079, 3288278229351791355200798719, 111414603535684224740921180159, 15370263527767281493147526365183

Offset: 1

K. D. Bajpai, May 04 2014

nonn

The n-th Woodall number is Wn = n*2^n - 1.

If Wn is semiprime, it is in the sequence.

			a(1) = 159 = (5*2^5 - 1) is 5th Woodall number and 159 = 3*53 which is semiprime.
a(2) = 895 = (7*2^7 - 1) is 7th Woodall number and 895 = 5*179 which is semiprime.

Amiram Eldar, Table of n, a(n) for n = 1..47 (terms 1..26 from K. D. Bajpai)

Cf. A001358, A002234, A003261, A050918, A242273.

with(numtheory): A242115:= proc(); if bigomega(x*2^x-1)=2 then RETURN (x*2^x-1); fi; end: seq(A242115 (),x=1..200);

Select[Table[n*2^n-1,{n,100}],PrimeOmega[#]==2&] (* Harvey P. Dale, Jan 03 2019 *)

for(n=1, 1000, if(bigomega(n*2^n-1)==2, print1(n*2^n-1, ", "))) \\ Colin Barker, May 07 2014

a(n) = A003261(A242273(n)). - Amiram Eldar, Nov 27 2019

A367561 Numbers k such that k^7*2^k - 1 is a prime.

Original entry on oeis.org

6, 45, 55, 80, 135, 187, 205, 384, 405, 1291, 1364, 2301, 2486, 2844, 16892, 27308, 30152, 32535, 45324, 71522, 72865

Offset: 1

Juri-Stepan Gerasimov, Nov 22 2023

No further terms <= 100000. - Michael S. Branicky, Aug 28 2024

Numbers k such that k^m*2^k - 1 is a prime: A000043 (m = 0), A002234 (m = 1), A058781 (m = 2), A367037 (m = 3), A367102 (m = 4), A367464 (m = 5), A367478 (m = 6), this sequence (m = 7).

Cf. A367560.

[k: k in [1..4000] | IsPrime(k^7*2^k-1)];

Select[Range[3000], PrimeQ[#^7*2^# - 1] &] (* Amiram Eldar, Nov 23 2023 *)

a(16)-a(21) from Michael S. Branicky, Nov 23 2023

A382646 Numbers k such that (k2^d - 1)(d*2^k - 1) is semiprime for some divisor d of k.

Original entry on oeis.org

2, 3, 6, 7, 12, 18, 19, 21, 30, 31, 42, 60, 75, 81, 115, 123, 126, 132, 133, 225, 249, 306, 324, 362, 384, 462, 468, 512, 606, 607, 612, 751, 822, 1279, 2170, 2202, 2281, 5312, 7755, 9531, 12379, 14898, 15822, 18123, 18819, 18885, 22971, 23005, 23208, 41628, 44497, 51384, 52540, 98726

Offset: 1

Juri-Stepan Gerasimov, Apr 01 2025

nonn

No further terms <= 10^5. - Michael S. Branicky, Apr 07 2025

			7 is in this sequence because (7*2^1-1)*(1*2^7-1) = 13*127 is semiprime for divisor 1 of 7.

Supersequence of A002234.

Cf. A001358, A003261, A382887.

[n: n in [1..1000] | not #[d: d in Divisors(n) | IsPrime(d*2^n-1) and IsPrime(n*2^d-1)] eq 0];

isok(k) = fordiv(k, d, if (ispseudoprime(k*2^d - 1) && ispseudoprime(d*2^k - 1), return(1))); \\ Michel Marcus, Apr 02 2025

from itertools import count, islice
from sympy import isprime, divisors
def A382646_gen(): # generator of terms
    yield from filter(lambda k:any(isprime((k<A382646_list = list(islice(A382646_gen(), 30)) # Chai Wah Wu, Apr 15 2025

a(40) from Michel Marcus, Apr 02 2025

a(41)-a(54) from Michael S. Branicky, Apr 07 2025

A099051 p*2^p - 1 where p is prime.

Original entry on oeis.org

7, 23, 159, 895, 22527, 106495, 2228223, 9961471, 192937983, 15569256447, 66571993087, 5085241278463, 90159953477631, 378231999954943, 6614661952700415, 477381560501272575, 34011184385901985791

Offset: 1

Parthasarathy Nambi, Nov 13 2004

This is the subset of Woodall numbers of prime index. The 9th largest known Woodall prime is in this sequence: 12379*2^12379-1, where 12379 is prime, as found by Wilfrid Keller in 1984. Smaller primes are when p = 2, 3, 751. These numbers can also be semiprime, as when p = 159, 163, or 211 and hard to factor as when n = 349 (108 digits). - Jonathan Vos Post, Nov 19 2004

			If p=3, 3*2^3 - 1 = 23.
If p=11, 11*2^11 - 1 = 22527.

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 360-361, 1996

Eric Weisstein's World of Mathematics, Woodall Numbers.

Similar to Woodall numbers (A003261). Cf. A002234.

Table[ Prime[n]*2^Prime[n] - 1, {n, 17}] (* Robert G. Wilson v, Nov 16 2004 *)

More terms from Robert G. Wilson v, Nov 15 2004

A137811 Number of digits in the n-th Woodall prime.

Original entry on oeis.org

1, 2, 3, 11, 25, 27, 37, 40, 78, 112, 119, 142, 157, 229, 251, 1603, 2339, 2874, 3731, 4768, 5690, 6920, 6930, 29725, 43058, 45468, 200815, 359799, 382007, 441847, 606279, 712818, 1129757, 5122515

Offset: 1

Ant King, Feb 12 2008

Woodall primes are prime numbers of the form k*2^k-1.

			As the sixth Woodall prime is a 27-digit number, we have a(6)= 27

Allan Cunningham and H. J. Woodall, Factorisation of Q=(2^q+-q) and (q 2^q+-1), Messenger Math., Vol. 47 (1917), pp. 1-38.
Wilfrid Keller, New Cullen Primes, Mathematics of Computation, Vol. 64, No. 212 (Ocober 1995), pp. 1733-1741.
Woodhall Primes, Definition And Status.

Cf. A055642, A050918, A137716, A002234.

IntegerLength/@Select[Table[n 2^n-1,{n,10000}],PrimeQ] (* The program generates the first 18 terms of the sequence. *) (* Harvey P. Dale, Feb 05 2023 *)

a(n) = A055642(A050918(n)).

a(28)-a(34) from Amiram Eldar, Jul 19 2025

cp's OEIS Frontend

A366898 Number of divisors of n*2^n - 1, the Woodall (or Riesel) numbers.

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A382811 Integers k such that d*2^k - 1 is prime for some divisor d of k.

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Magma

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A367464 Numbers k such that k^5*2^k - 1 is a prime.

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A367478 Numbers k such that k^6*2^k - 1 is a prime.

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Magma

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A210340 Generalized Woodall primes: any primes that can be written in the form n*b^n - 1 with n+2 > b > 2.

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A242115 Woodall semiprimes: Semiprimes of the form n*2^n - 1.

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A367561 Numbers k such that k^7*2^k - 1 is a prime.

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Magma

Mathematica

Extensions

A382646 Numbers k such that (k*2^d - 1)*(d*2^k - 1) is semiprime for some divisor d of k.

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A382646 Numbers k such that (k2^d - 1)(d*2^k - 1) is semiprime for some divisor d of k.