A003261 -id:A003261 - cp's OEIS frontend

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A180247 Prime Brier numbers: primes p such that for all k >= 1 the numbers p2^k + 1 and p2^k - 1 are composite.

10439679896374780276373, 21444598169181578466233, 105404490005793363299729, 178328409866851219182953, 239365215362656954573813, 378418904967987321998467, 422280395899865397194393, 474362792344501650476113, 490393518369132405769309

Offset: 1

Arkadiusz Wesolowski, Aug 19 2010

nonn

WARNING: These are just the smallest examples known - there may be smaller ones. Even the first term is uncertain. - N. J. A. Sloane, Jun 20 2017
There are no prime Brier numbers below 10^10. - Arkadiusz Wesolowski, Jan 12 2011
It is a conjecture that every such number has more than 11 digits. In 2011 I have calculated that for any prime p < 10^11 there is a k such that either p*2^k + 1 or p*2^k - 1 has all its prime factors greater than 1321. - Arkadiusz Wesolowski, Feb 03 2016
The first term was found by Dan Ismailescu and Peter Seho Park and the next two by Christophe Clavier (see below). See also A076335. - N. J. A. Sloane, Jan 03 2014
a(4)-a(9) computed in 2017 by the author.

D. Baczkowski, J. Eitner, C. E. Finch, B. Suminski, and M. Kozek, Polygonal, Sierpinski, and Riesel numbers, Journal of Integer Sequences, 2015 Vol 18. #15.8.1.
Chris Caldwell, The Prime Glossary, Riesel number
Chris Caldwell, The Prime Glossary, Sierpinski number
Christophe Clavier, 14 new Brier numbers
Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975), pp. 79-81.
P. Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), pp. 113-123.
Yves Gallot, A search for some small Brier numbers, 2000.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 6992565235279559197457863
Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, 16 (2013), #13.9.8.
Joe McLean, Brier Numbers [Cached copy]
Carlos Rivera, Problem 52. ±p ± 2^n, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Brier Number

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194638, A194639, A076336, A076337, A040081, A040076, A103963, A103964, A038699, A050921, A064699, A052333, A003261.

These are the primes in A076335.

Entry revised by N. J. A. Sloane, Jan 03 2014

Entry revised by Arkadiusz Wesolowski, May 29 2017

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

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

A099035 a(n) = (n+1)*2^(n-1) - 1.

Original entry on oeis.org

1, 5, 15, 39, 95, 223, 511, 1151, 2559, 5631, 12287, 26623, 57343, 122879, 262143, 557055, 1179647, 2490367, 5242879, 11010047, 23068671, 48234495, 100663295, 209715199, 436207615, 905969663, 1879048191, 3892314111, 8053063679

Offset: 1

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Author

Ralf Stephan, Sep 28 2004

Keywords

nonn
easy

Comments

Row sums of triangle A135852. - Gary W. Adamson, Dec 01 2007

Binomial transform of [1, 4, 6, 8, 10, 12, 14, 16, ...]. Equals A128064 * A000225, (A000225 starting 1, 3, 7, 15, ...). - Gary W. Adamson, Dec 28 2007

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Crossrefs

First differences of A066524.

Cf. A135852, A128064, A000225.

Programs

Magma

[(n+1)*2^(n-1) -1: n in [1..30]]; // G. C. Greubel, Dec 31 2017

Mathematica

Table[(n + 1)*2^(n - 1) - 1, {n,1,30}] (* G. C. Greubel, Dec 31 2017 *) LinearRecurrence[{5,-8,4},{1,5,15},30] (* Harvey P. Dale, Dec 28 2022 *)

PARI

a(n)=(n+1)*2^(n-1)-1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = A057711(n+1) - 1 = A058966(n+3)/2 = (A087323(n)-1)/2 = (A074494(n+1)-2)/3 = (A003261(n+1)-3)/4 = A036289(n+1)/4 - 1, n>0.

a(n) = A131056(n+1) - 2. - Juri-Stepan Gerasimov, Oct 02 2011

From Colin Barker, Mar 23 2012: (Start)

a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3).

G.f.: x*(1-2*x^2)/((1-x)*(1-2*x)^2). (End)

E.g.f.: ((2*x+1)*exp(2*x) - 2*exp(x) + 1)/2. - G. C. Greubel, Dec 31 2017

A056041 Value for which b(a(n))=0 when b(2)=n and b(k+1) is calculated by writing b(k) in base k, reading this as being written in base k+1 and then subtracting 1.

Original entry on oeis.org

2, 3, 5, 7, 23, 63, 383, 2047

Offset: 0

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Author

Henry Bottomley, Aug 04 2000

Keywords

base
nonn

Comments

a(8)=3*2^(3*2^27+27)-1 which is more than 10^(10^8) and equal to the final base of the Goodstein sequence starting with g(2)=4; indeed, apart from the initial term, the sequence starting with b(2)=8 is identical to the Goodstein sequence starting with g(2)=4. The initial terms of a(n) [2, 3, 5 and 7] are equal to the initial terms of the equivalent final bases of Goodstein sequences starting at the same points. a(9)=2^(2^(2^70+70)+2^70+70)-1 which is more than 10^(10^(10^20)).

It appears that if n is even then a(n) is one less than three times a power of two, while if n is odd then a(n) is one less than a power of two.

Comment from John Tromp, Dec 02 2004: The sequence 2,3,5,7,3*2^402653211 - 1, ... gives the final base of the Goodstein sequence starting with n. This is an example of a very rapidly growing function that is total (i.e. defined on any input), although this fact is not provable in first-order Peano Arithmetic. See the links for definitions. This grows even faster than the Friedman sequence described in the Comments to A014221.

In fact there are two related sequences: (i) The Goodstein function l(n) = number of steps for the Goodstein sequence to reach 0 when started with initial term n >= 0: 0, 1, 3, 5, 3*2^402653211 - 3, ...; and (ii) the same sequence + 2: 2, 3, 5, 7, 3*2^402653211 - 1, ..., which is the final base reached. Both grow too rapidly to have their own entries in the database.

Related to the hereditary base sequences - see cross-reference lines.

This sequence gives the final base of the weak Goodstein sequence starting with n; compare A266203, the length of the weak Goodstein sequence. a(n) = A266203(n) + 2.

Examples

a(3)=7 because starting with b(2)=3=11 base 2, we get b(3)=11-1 base 3=10 base 3=3, b(4)=10-1 base 4=3, b(5)=3-1 base 5=2, b(6)=2-1 base 6=1 and b(7)=1-1 base 7=0.

Links

R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.

L. Kirby, and J. Paris, Accessible independence results for Peano arithmetic, Bull. London Mathematical Society, 14 (1982), 285-293.

J. Tromp, Programming Pearls

Eric Weisstein's World of Mathematics, Goodstein Sequence

Wikipedia, Goodstein's theorem

Crossrefs

Cf. A266202, A268687, A268689, A268688.

Equals A266203 + 2.

Weak Goodstein sequences: A267647, A267648, A271987, A271988, A271989, A271990, A271991, A137411, A271992, A265034.

Steps of strong Goodstein sequences: A056004, A057650, A059934, A059935, A059936, A271977.

Strong Goodstein sequences: A215409, A056193, A266204, A222117, A059933.

Woodall numbers: A003261.

A060416 a(n) = n*4^n - 1, with a(0) = 1.

Original entry on oeis.org

1, 3, 31, 191, 1023, 5119, 24575, 114687, 524287, 2359295, 10485759, 46137343, 201326591, 872415231, 3758096383, 16106127359, 68719476735, 292057776127, 1236950581247, 5222680231935, 21990232555519, 92358976733183, 387028092977151, 1618481116086271, 6755399441055743

Offset: 0

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Author

Jason Earls, Apr 05 2001

Keywords

nonn
easy

Links

Harry J. Smith, Table of n, a(n) for n = 0..200

Paul Leyland, Factors of Cullen and Woodall numbers.

Paul Leyland, Generalized Cullen and Woodall numbers.

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.

Index entries for linear recurrences with constant coefficients, signature (9,-24,16).

Crossrefs

Cf. A003261, A018215.

Programs

Mathematica

CoefficientList[Series[(1 - 6 x + 28 x^2 - 32 x^3)/((1 - x) (1 - 4 x)^2), {x, 0, 21}], x] (* Michael De Vlieger, Jan 04 2020 *)

PARI

a(n) = { abs(n*4^n - 1) } \\ Harry J. Smith, Jul 04 2009

Formula

G.f.: (1 - 6*x + 28*x^2 - 32*x^3)/((1 - x)*(1 - 4*x)^2). - Colin Barker, Apr 22 2012

From Elmo R. Oliveira, May 05 2025: (Start)

E.g.f.: 2 + exp(x)*(4*x*exp(3*x) - 1).

a(n) = A018215(n) - 1 for n >= 1.

a(n) = 9*a(n-1) - 24*a(n-2) + 16*a(n-3) for n > 3. (End)

A064751 a(n) = n*5^n - 1.

Original entry on oeis.org

4, 49, 374, 2499, 15624, 93749, 546874, 3124999, 17578124, 97656249, 537109374, 2929687499, 15869140624, 85449218749, 457763671874, 2441406249999, 12969970703124, 68664550781249, 362396240234374, 1907348632812499, 10013580322265624, 52452087402343749, 274181365966796874

Offset: 1

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Author

N. J. A. Sloane, Oct 19 2001

Keywords

nonn
easy

Links

Harry J. Smith, Table of n, a(n) for n = 1..150

Paul Leyland, Factors of Cullen and Woodall numbers.

Paul Leyland, Generalized Cullen and Woodall numbers.

Index entries for linear recurrences with constant coefficients, signature (11,-35,25).

Crossrefs

Cf. A003261, A036291.

Programs

Magma

[n*5^n - 1: n in [1..30]]; // Vincenzo Librandi, Jun 21 2018

Mathematica

Table[n*5^n-1,{n,20}] (* or *) LinearRecurrence[{11,-35,25},{4,49,374},20] (* Harvey P. Dale, Jun 25 2017 *) CoefficientList[Series[(4 + 5 x - 25 x^2) / ((1 - 5 x)^2 (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 21 2018 *)

PARI

a(n) = { n*5^n - 1 } \\ Harry J. Smith, Sep 24 2009

Formula

G.f.: x*(4 + 5*x - 25*x^2)/((1 - 5*x)^2*(1 - x)). - Vincenzo Librandi, Jun 21 2018

a(n) = A036291(n) - 1. - Michel Marcus, Jun 21 2018

From Elmo R. Oliveira, May 05 2025: (Start)

E.g.f.: 1 + exp(x)*(5*x*exp(4*x) - 1).

a(n) = 11*a(n-1) - 35*a(n-2) + 25*a(n-3) for n > 3. (End)

A064752 a(n) = n*6^n - 1.

Original entry on oeis.org

5, 71, 647, 5183, 38879, 279935, 1959551, 13436927, 90699263, 604661759, 3990767615, 26121388031, 169789022207, 1097098297343, 7052774768639, 45137758519295, 287753210560511, 1828079220031487, 11577835060199423, 73123168801259519, 460675963447934975, 2895677484529876991

Offset: 1

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Author

N. J. A. Sloane, Oct 19 2001

Keywords

nonn
easy

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Paul Leyland, Factors of Cullen and Woodall numbers.

Paul Leyland, Generalized Cullen and Woodall numbers.

Index entries for linear recurrences with constant coefficients, signature (13,-48,36).

Crossrefs

Cf. A003261, A036292.

Programs

Magma

[ n*6^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011

Mathematica

Table[n 6^n-1,{n,25}] (* or *) LinearRecurrence[{13,-48,36},{5,71,647}, 25] (* Harvey P. Dale, May 25 2011 *)

Formula

From Harvey P. Dale, May 25 2011: (Start)

a(n) = 13*a(n-1) - 48*a(n-2) + 36*a(n-3); a(1)=5, a(2)=71, a(3)=647.

G.f.: 6*x/(1-6*x)^2 + x/(x-1). (End)

From Elmo R. Oliveira, May 05 2025: (Start)

E.g.f.: 1 + exp(x)*(6*x*exp(5*x) - 1).

a(n) = A036292(n) - 1. (End)

A064754 a(n) = n*8^n - 1.

Original entry on oeis.org

7, 127, 1535, 16383, 163839, 1572863, 14680063, 134217727, 1207959551, 10737418239, 94489280511, 824633720831, 7146825580543, 61572651155455, 527765581332479, 4503599627370495, 38280596832649215, 324259173170675711, 2738188573441261567, 23058430092136939519

Offset: 1

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Author

N. J. A. Sloane, Oct 19 2001

Keywords

nonn
easy

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Paul Leyland, Factors of Cullen and Woodall numbers.

Paul Leyland, Generalized Cullen and Woodall numbers.

Index entries for linear recurrences with constant coefficients, signature (17,-80,64).

Crossrefs

Cf. A003261, A036294.

Programs

Magma

[ n*8^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011

Mathematica

Table[n*8^n-1,{n,20}] (* or *) LinearRecurrence[{17,-80,64},{7,127,1535},20] (* Harvey P. Dale, May 20 2013 *)

Formula

G.f.: x*(64*x^2 - 8*x - 7)/((x-1)*(8*x-1)^2). - Colin Barker, Oct 15 2012

a(n) = 17*a(n-1) - 80*a(n-2) + 64*a(n-3); a(1)=7, a(2)=127, a(3)=1535. - Harvey P. Dale, May 20 2013

From Elmo R. Oliveira, May 05 2025: (Start)

E.g.f.: 1 + exp(x)*(8*x*exp(7*x) - 1).

a(n) = A036294(n) - 1. (End)

A064755 a(n) = n*9^n - 1.

Original entry on oeis.org

8, 161, 2186, 26243, 295244, 3188645, 33480782, 344373767, 3486784400, 34867844009, 345191655698, 3389154437771, 33044255768276, 320275094369453, 3088366981419734, 29648323021629455, 283512088894331672, 2701703435345984177, 25666182635786849690, 243153309181138576019

Offset: 1

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Author

N. J. A. Sloane, Oct 19 2001

Keywords

nonn
easy

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Paul Leyland, Factors of Cullen and Woodall numbers.

Paul Leyland, Generalized Cullen and Woodall numbers.

Index entries for linear recurrences with constant coefficients, signature (19,-99,81).

Crossrefs

Cf. A003261, A158749.

Programs

Magma

[ n*9^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011

Mathematica

Table[n*9^n-1,{n,20}] (* Harvey P. Dale, Feb 28 2017 *)

Formula

From Chai Wah Wu, Feb 01 2018: (Start)

a(n) = 19*a(n-1) - 99*a(n-2) + 81*a(n-3) for n > 3.

G.f.: x*(81*x^2 - 9*x - 8)/((x - 1)*(9*x - 1)^2). (End)

From Elmo R. Oliveira, May 05 2025: (Start)

E.g.f.: 1 + exp(x)*(9*x*exp(8*x) - 1).

a(n) = A158749(n) - 1. (End)

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

A180247 Prime Brier numbers: primes p such that for all k >= 1 the numbers p*2^k + 1 and p*2^k - 1 are composite.

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Comments

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Crossrefs

Extensions

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

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Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

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Comments

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Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A099035 a(n) = (n+1)*2^(n-1) - 1.

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A056041 Value for which b(a(n))=0 when b(2)=n and b(k+1) is calculated by writing b(k) in base k, reading this as being written in base k+1 and then subtracting 1.

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Crossrefs

A060416 a(n) = n*4^n - 1, with a(0) = 1.

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Crossrefs

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Mathematica

PARI

Formula

A064751 a(n) = n*5^n - 1.

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Programs

Magma

Mathematica

PARI

Formula

A064752 a(n) = n*6^n - 1.

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A180247 Prime Brier numbers: primes p such that for all k >= 1 the numbers p2^k + 1 and p2^k - 1 are composite.