A002234 -id:A002234 - cp's OEIS frontend

A003261 Woodall (or Riesel) numbers: n*2^n - 1.

1, 7, 23, 63, 159, 383, 895, 2047, 4607, 10239, 22527, 49151, 106495, 229375, 491519, 1048575, 2228223, 4718591, 9961471, 20971519, 44040191, 92274687, 192937983, 402653183, 838860799, 1744830463, 3623878655, 7516192767, 15569256447, 32212254719, 66571993087

Offset: 1

N. J. A. Sloane

For n>1, a(n) is base at which zero is reached for the function "write f(j) in base j, read as base j+1 and then subtract 1 to give f(j+1)" starting from f(n) = n^2 - 1. - Henry Bottomley, Aug 06 2000

Sequence corresponds also to the maximum chain length of the classic puzzle whereby, under agreed commercial terms, an asset of unringed golden chain, when judiciously fragmented into as few as n pieces and n-1 opened links (through n-1 cuts), might be used to settle debt sequentially, with a golden link covering for unit cost. Here beside the n-1 opened links, the n fragmented pieces have lengths n, 2*n, 4*n, ..., 2^(n-1)*n. For instance, the chain of original length a(5)=159, if segregated by 4 cuts into 5+1+10+1+20+1+40+1+80, may be used to pay sequentially, i.e., a link-cost at a time, for an equivalent cost up to 159 links, to the same creditor. - Lekraj Beedassy, Feb 06 2003

			G.f. = x + 7*x^2 + 23*x^3 + 63*x^4 + 159*x^5 + 383*x^6 + 895*x^7 + ... - _Michael Somos_, Nov 04 2018

A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 159.
K. R. Bhutani and A. B. Levin, "The Problem of Sawing a Chain", Journal of Recreational Mathematics 2002-3 31(1) 32-35.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
M. Gardner, Martin Gardner's Sixth Book of Mathematical Diversions from Scientific American, "Gold Links", Problem 4, pp. 50-51; 57-58, University of Chicago Press, 1983.
O. O'Shea, Mathematical Brainteasers with Surprising Solutions, Problem 76, pp. 183-185, Prometheus Books, Guilford, Connecticut, 2020.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vladimir Pletser, Table of n, a(n) for n = 1..3000 (terms 1..300 from T. D. Noe).
Ray Ballinger, Woodall Primes: Definition and Status.
Attila Bérczes, István Pink, and Paul Thomas Young, Cullen numbers and Woodall numbers in generalized Fibonacci sequences, J. Num. Theor. (2024) Vol. 262, 86-102.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 159.
C. K. Caldwell, Woodall Numbers.
Orhan Eren and Yüksel Soykan, Gaussian Generalized Woodall Numbers, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
Orhan Eren and Yüksel Soykan, On Dual Hyperbolic Generalized Woodall Numbers, Arch. Current Res. Int'l (2024) Vol. 24, Iss. 11, Art. No. ACRI.126420, 398-423. See p. 401.
Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 27.
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
D. Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, 17 (2014), #14.9.4.
Hisanori Mishima, Factorizations of many number sequences: Riesel numbers, n=1..100, n=101..200, n=201..300, n=301..323.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
T. Sillke, Using Chains Links To Pay For A Room.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Woodall Number.
Wikipedia, Woodall number.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A002234, A002064, A005849, A050918, A006127.
a(n) = A036289(n) - 1 = A002064(n) - 2.
Cf. A133653.

a003261 = (subtract 1) . a036289  -- Reinhard Zumkeller, Mar 05 2012

[n*2^n - 1: n in [1..30]]; // G. C. Greubel, Nov 04 2018

for n from 1 to 3000 do n, n*2^n -1; end do; # Vladimir Pletser, Dec 30 2022

Table[n*2^n-1,{n,3*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
LinearRecurrence[{5,-8,4},{1,7,23},30] (* Harvey P. Dale, Mar 13 2022 *)

A003261(n)=n*2^n-1  \\ M. F. Hasler, Oct 31 2012

[n*2**n - 1 for n in range(1, 29)] # Michael S. Branicky, Jan 07 2021

G.f.: x*(-1-2*x+4*x^2) / ( (x-1)*(-1+2*x)^2 ). - Simon Plouffe in his 1992 dissertation
Binomial transform of A133653 and double binomial transform of [1, 5, -1, 1, -1, 1, ...]. - Gary W. Adamson, Sep 19 2007
a(n) = -(2)^n * A006127(-n) for all n in Z. - Michael Somos, Nov 04 2018
E.g.f.: 1 + exp(x)*(2*exp(x)*x - 1). - Stefano Spezia, Nov 24 2024

A005849 Indices of prime Cullen numbers: numbers k such that k*2^k + 1 is prime.

Original entry on oeis.org

1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881

Offset: 1

N. J. A. Sloane

From Amiram Eldar, Jun 05 2021: (Start)
The terms were found by:
a(1) - Cullen (1905). He found that there are no other terms up to 100 with the possible exception of 53. Cunningham (1906) showed that the 53rd Cullen number is composite and that the only possible term up to 200 is 141.
a(2) - Robinson (1958).
a(3)-a(6) - Keller (1995).
a(7)-a(8) - Masakatu Morii (1997).
a(9)-a(10) - Jeffrey Young (1997).
a(11)-a(12) - Darren Smith (1998).
a(13) - Masakatu Morii (1998).
a(14) - Mark Rodenkirch (2005).
a(15) - Dennis R. Gesker (2009).
a(16) - Magnus Bergman (2009). (End)

A. J. Cunningham, Solution of question 15897, Math. Quest. Educ. Times, Vol. 10 (1906), pp. 44-47.
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 141, p. 48, Ellipses, Paris 2008.
Harvey Dubner, Generalized Cullen numbers, J. Rec. Math., Vol. 21, No. 3 (1989), pp. 190-191.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B20.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 283.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Ballinger, Cullen Primes: Definition and Status.
Chris K. Caldwell, The Top Twenty: Cullen Primes.
Allan Cunningham and H. J. Woodall, Factorisation of Q = (2^q -/+ q) and (q*2^q -\+ 1), The Messenger of Mathematics, Vol. 47 (1917-18), pp. 1-38. See p. 22.
Harvey Dubner, Generalized Cullen numbers, J. Rec. Math., Vol. 21, No. 3 (1989), pp. 190-191. (Annotated scanned copy)
Wilfrid Keller, New Cullen primes, Mathematics of Computation, Vol. 64, No. 212 (1995), pp. 1733-1741.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
PrimeGrid, PrimeGrid Primes: Subproject: (CUL) Cullen Prime Search.
Raphael M. Robinson, A Report on primes of the form k*2^n + 1 and on factors of Fermat numbers, Proceedings of the American Mathematical Society, Vol. 9, No. 5 (1958), pp. 673-681.
Eric Weisstein's World of Mathematics, Cullen Number.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Cf. A002064, A002234, A050920, A173474 (complement).

Select[Range[1000], PrimeQ[# 2^# + 1] &] (* Alonso del Arte, Jul 30 2017 *)

is(n)=isprime(n<Charles R Greathouse IV, Feb 06 2017

a(14) = 1354828 from old Proth Search pages by Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Apr 20 2006
The term 1467763 was added in error and has now been deleted; Jens Kruse Andersen, Nov 28 2007, remarks that 1467763 * 2^1467763 - 1 is a Woodall prime, but 3 divides the Cullen number 1467763 * 2^1467763 + 1.
6328548 from John Blazek, May 14 2009. He later reports that the search of the range from 6300000 to 6328548 was completed on May 28 2009.
Added a(16) = 6679881 from Caldwell's page, fixed broken link. - M. F. Hasler, Jan 18 2015
Name edited by Andrey Zabolotskiy and Felix Fröhlich, May 28 2021

A050918 Woodall primes: primes of form k*2^k-1.

Original entry on oeis.org

7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319, 1307960347852357218937346147315859062783, 225251798594466661409915431774713195745814267044878909733007331390393510002687

Offset: 1

N. J. A. Sloane, Dec 30 1999

M. F. Hasler, Table of n, a(n) for n = 1..15 (all terms < 10^999).
Ray Ballinger, Woodall Primes: Definition and Status.
C. K. Caldwell, Woodall Numbers.
Brady Haran and Matt Parker, 383 is cool, Numberphile video, 2017.
Eric Weisstein's World of Mathematics, Woodall Number.

Cf. A002234, A003261.

Select[Table[n 2^n - 1, {n, 300}], PrimeQ] (* Harvey P. Dale, Jul 12 2012 *)

for(n=2,999,ispseudoprime(p=n*2^n-1)&&print1(p",")) \\ M. F. Hasler, May 10 2017

from sympy import isprime
def auptok(limit):
    return list(filter(isprime, (k*2**k-1 for k in range(1, limit+1))))
print(auptok(1000)) # Michael S. Branicky, Jul 23 2021

a(n) = A002234(n)*2^A002234(n) - 1. - M. F. Hasler, May 10 2017

A086661 Numbers k such that k*4^k-1 is prime.

Original entry on oeis.org

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

Offset: 1

Farideh Firoozbakht, Jul 27 2003

2, 3, 5, 23, 107, 1973, 20747 is the subsequence of prime terms.

			2 is in the sequence because 2*4^2-1=31 is prime.
3 is in the sequence because 3*4^3-1=191 is prime.

H. Dubner, Generalized Cullen Numbers, J. Rec. Math, 21 (No. 3, 1989).

Steven Harvey, Generalized Woodall Search

Cf. A007646, A002234, A005849.

Do[If[PrimeQ[n*4^n-1], Print[n]], {n, 4000}]
Select[Range[2000],PrimeQ[# 4^#-1]&] (* Harvey P. Dale, Nov 09 2024 *)

is(n)=ispseudoprime(n*4^n-1) \\ Charles R Greathouse IV, May 22 2017

One more term from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Nov 23 2004
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
Prepended first terms 1 and 2 - Pierre CAMI, Jul 21 2014
a(20)-a(21) from Harvey link by Ray Chandler, Apr 10 2016

A242200 Numbers n such that n*7^n - 1 is prime.

Original entry on oeis.org

2, 18, 68, 84, 3812, 14838, 51582

Offset: 1

Vincenzo Librandi, May 09 2014

Steven Harvey, Generalized Woodall Search

Cf. numbers n such that n*k^n - 1 is prime. A002234 (k=2), A006553 (k=3), A086661 (k=4), A059676 (k=5), A059675 (k=6), this sequence (k=7), A242201 (k=8), A242202 (k=9), A059671 (k=10).

[n: n in [0..3000] | IsPrime(n*7^n-1)];

Select[Range[2000], PrimeQ[# 7^# - 1] &]

is(n)=ispseudoprime(n*7^n-1) \\ Charles R Greathouse IV, May 22 2017

a(5) - a(7) from Harvey's list (see Links).

A242273 Numbers n such that n*2^n - 1 is a semiprime.

Original entry on oeis.org

5, 7, 8, 9, 10, 12, 18, 20, 25, 32, 37, 39, 72, 80, 85, 90, 97, 142, 150, 159, 163, 168, 169, 186, 192, 211, 231, 272, 305, 349, 363, 369, 375, 463, 465, 615, 668, 672, 789, 797, 817, 859, 908, 938, 951, 1092, 1123

Offset: 1

Vincenzo Librandi, May 12 2014

The semiprimes of this form are: 159, 895, 2047, 4607, 10239, ... (A242115).

a(48) >= 1152. - Hugo Pfoertner, Jul 29 2019

FactorDB, Status of 1152*2^1152-1.

Cf. numbers n such that n*k^n - 1 is semiprime: this sequence (k=2), A242274 (k=3), A242335 (k=4), A242336 (k=5), A242337 (k=6), A242338 (k=7), A242339 (k=8), A242340 (k=9), A242341 (k=10).

Cf. A002234, A003261, A242115, A242175.

IsSemiprime:=func; [n: n in [2..1000] | IsSemiprime(s) where s is n*2^n-1];

Mathematica

Select[Range[1000], PrimeOmega[# 2^# - 1]==2&]

Formula

A003261(a(n)) = A242115(n). - Amiram Eldar, Nov 27 2019

Extensions

a(28)-a(29) from Luke March, Aug 05 2015

a(30)-a(42) from Carl Schildkraut, Aug 18 2015

Corrected and extended by Luke March, Sep 01 2015

Missing terms a(26)-a(27) inserted by Amiram Eldar, Nov 27 2019

cp's OEIS Frontend

A172297 Partial sums of A002234.

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A003261 Woodall (or Riesel) numbers: n*2^n - 1.

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A005849 Indices of prime Cullen numbers: numbers k such that k*2^k + 1 is prime.

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A050918 Woodall primes: primes of form k*2^k-1.

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A086661 Numbers k such that k*4^k-1 is prime.

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A242200 Numbers n such that n*7^n - 1 is prime.

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A242273 Numbers n such that n*2^n - 1 is a semiprime.

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