cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A000043 Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Equivalently, integers k such that 2^k - 1 is prime.
It is believed (but unproved) that this sequence is infinite. The data suggest that the number of terms up to exponent N is roughly K log N for some constant K.
Length of prime repunits in base 2.
The associated perfect number N=2^(p-1)*M(p) (=A019279*A000668=A000396), has 2p (=A061645) divisors with harmonic mean p (and geometric mean sqrt(N)). - Lekraj Beedassy, Aug 21 2004
In one of his first publications Euler found the numbers up to 31 but erroneously included 41 and 47.
Equals number of bits in binary expansion of n-th Mersenne prime (A117293). - Artur Jasinski, Feb 09 2007
Number of divisors of n-th even perfect number, divided by 2. Number of divisors of n-th even perfect number that are powers of 2. Number of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 24 2008
Number of divisors of n-th even superperfect number A061652(n). Numbers of divisors of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Mar 01 2008
Differences between exponents when the even perfect numbers are represented as differences of powers of 2, for example: The 5th even perfect number is 33550336 = 2^25 - 2^12 then a(5)=25-12=13 (see A135655, A133033, A090748). - Omar E. Pol, Mar 01 2008
Number of 1's in binary expansion of n-th even perfect number (see A135650). Number of 1's in binary expansion of divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n) (see A135652, A135653, A135654, A135655). - Omar E. Pol, May 04 2008
Indices of the numbers A006516 that are also even perfect numbers. - Omar E. Pol, Aug 30 2008
Indices of Mersenne numbers A000225 that are also Mersenne primes A000668. - Omar E. Pol, Aug 31 2008
The (prime) number p appears in this sequence if and only if there is no prime q<2^p-1 such that the order of 2 modulo q equals p; a special case is that if p=4k+3 is prime and also q=2p+1 is prime then the order of 2 modulo q is p so p is not a term of this sequence. - Joerg Arndt, Jan 16 2011
Primes p such that sigma(2^p) - sigma(2^p-1) = 2^p-1. - Jaroslav Krizek, Aug 02 2013
Integers k such that every degree k irreducible polynomial over GF(2) is also primitive, i.e., has order 2^k-1. Equivalently, the integers k such that A001037(k) = A011260(k). - Geoffrey Critzer, Dec 08 2019
Conjecture: for k > 1, 2^k-1 is (a Mersenne) prime or k = 2^(2^m)+1 (is a Fermat number) if and only if (k-1)^(2^k-2) == 1 (mod (2^k-1)k^2). - Thomas Ordowski, Oct 05 2023
Conjecture: for p prime, 2^p-1 is (a Mersenne) prime or p = 2^(2^m)+1 (is a Fermat number) if and only if (p-1)^(2^p-2) == 1 (mod 2^p-1). - David Barina, Nov 25 2024
Already as of Dec. 2020, all exponents up to 10^8 had been verified, implying that 74207281, 77232917 and 82589933 are indeed the next three terms. As of today, all exponents up to 130439863 have been tested at least once, see the GIMPS Milestones Report. - M. F. Hasler, Apr 11 2025
On June 23. 2025 all exponents up to 74340751 have been verified, confirming that 74207281 is the exponent of the 49th Mersenne Prime. - Rodolfo Ruiz-Huidobro, Jun 23 2025

Examples

			Corresponding to the initial terms 2, 3, 5, 7, 13, 17, 19, 31 ... we get the Mersenne primes 2^2 - 1 = 3, 2^3 - 1 = 7, 2^5 - 1 = 31, 127, 8191, 131071, 524287, 2147483647, ... (see A000668).

References

  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 4.
  • J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, p. 79.
  • R. K. Guy, Unsolved Problems in Number Theory, Section A3.
  • F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 57.
  • Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 19.
  • Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 132-134.
  • B. Tuckerman, The 24th Mersenne prime, Notices Amer. Math. Soc., 18 (Jun, 1971), Abstract 684-A15, p. 608.

Links

Crossrefs

Cf. A000668 (Mersenne primes).
Cf. A028335 (integer lengths of Mersenne primes).
Cf. A000225 (Mersenne numbers).
Cf. A001348 (Mersenne numbers with n prime).
Cf. A016027, A046051, A057429, A057951-A057958, A066408, A117293, A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A134458, A000225, A000396, A090748, A133033, A135655, A006516, A019279, A061652, A133033, A135650, A135652, A135653, A135654, A260073, A050475, A379590.

Programs

  • Mathematica
    MersennePrimeExponent[Range[48]] (* Eric W. Weisstein, Jul 17 2017; updated Oct 21 2024 *)
  • PARI
    isA000043(n) = isprime(2^n-1) \\ Michael B. Porter, Oct 28 2009
  • PARI
    is(n)=my(h=Mod(2,2^n-1)); for(i=1, n-2, h=2*h^2-1); h==0||n==2 \\ Lucas-Lehmer test for exponent e. - Joerg Arndt, Jan 16 2011, and Charles R Greathouse IV, Jun 05 2013
forprime(e=2,5000,if(is(e),print1(e,", "))); /* terms < 5000 */
  • Python
    from sympy import isprime, prime
for n in range(1,100):
    if isprime(2**prime(n)-1):
        print(prime(n), end=', ') # Stefano Spezia, Dec 06 2018

Formula

a(n) = log((1/2)*(1+sqrt(1+8*A000396(n))))/log(2). - Artur Jasinski, Sep 23 2008 (under the assumption there are no odd perfect numbers, Joerg Arndt, Feb 23 2014)
a(n) = A000005(A061652(n)). - Omar E. Pol, Aug 26 2009
a(n) = A000120(A000396(n)), assuming there are no odd perfect numbers. - Omar E. Pol, Oct 30 2013

Extensions

Also in the sequence: p = 74207281. - Charles R Greathouse IV, Jan 19 2016
Also in the sequence: p = 77232917. - Eric W. Weisstein, Jan 03 2018
Also in the sequence: p = 82589933. - Gord Palameta, Dec 21 2018
a(46) = 42643801 and a(47) = 43112609, whose ordinal positions in the sequence are now confirmed, communicated by Eric W. Weisstein, Apr 12 2018
a(48) = 57885161, whose ordinal position in the sequence is now confirmed, communicated by Benjamin Przybocki, Jan 05 2022
Also in the sequence: p = 136279841. - Eric W. Weisstein, Oct 21 2024
As of Jan 31 2025, 48 terms are known, and are shown in the DATA section. Four additional numbers are known to be in the sequence, namely 74207281, 77232917, 82589933, and 136279841, but they may not be the next terms. See the GIMP website for the latest information. - N. J. A. Sloane, Jan 31 2025

A007583 a(n) = (2^(2*n + 1) + 1)/3.

Original entry on oeis.org

1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763, 699051, 2796203, 11184811, 44739243, 178956971, 715827883, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 2932031007403, 11728124029611, 46912496118443, 187649984473771, 750599937895083
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

Let u(k), v(k), w(k) be the 3 sequences defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)-w(k), v(k+1)=u(k)-v(k)+w(k), w(k+1)=-u(k)+v(k)+w(k); let M(k)=Max(u(k),v(k),w(k)); then a(n)=M(2n)=M(2n-1). - Benoit Cloitre, Mar 25 2002
Also the number of words of length 2n generated by the two letters s and t that reduce to the identity 1 by using the relations ssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the dihedral group D6=C2xD3. - Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com) and John W. Layman, Jul 08 2002
Binomial transform of A025192. - Paul Barry, Apr 11 2003
Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_6. Example: a(1)=3 because in the cycle ABCDEF we have three walks of length 3 between A and B: ABAB, ABCB and AFAB. - Emeric Deutsch, Apr 01 2004
Numbers of the form 1 + Sum_{i=1..m} 2^(2*i-1). - Artur Jasinski, Feb 09 2007
Prime numbers of the form 1+Sum[2^(2n-1)] are in A000979. Numbers x such that 1+Sum[2^(2n-1)] is prime for n=1,2,...,x is A127936. - Artur Jasinski, Feb 09 2007
Related to A024493(6n+1), A131708(6n+3), A024495(6n+5). - Paul Curtz, Mar 27 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-6, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). - Milan Janjic, Feb 21 2010
Number of toothpicks in the toothpick structure of A139250 after 2^n stages. - Omar E. Pol, Feb 28 2011
Numbers whose binary representation is "10" repeated (n-1) times with "11" appended on the end, n >= 1. For example 171 = 10101011 (2). - Omar E. Pol, Nov 22 2012
a(n) is the smallest number for which A072219(a(n)) = 2*n+1. - Ramasamy Chandramouli, Dec 22 2012
An Engel expansion of 2 to the base b := 4/3 as defined in A181565, with the associated series expansion 2 = b + b^2/3 + b^3/(3*11) + b^4/(3*11*43) + .... Cf. A007051. - Peter Bala, Oct 29 2013
The positive integer solution (x,y) of 3*x - 2^n*y = 1, n>=0, with smallest x is (a(n/2), 2) if n is even and (a((n-1)/2), 1) if n is odd. - Wolfdieter Lang, Feb 15 2014
The smallest positive number that requires at least n additions and subtractions of powers of 2 to be formed. See Puzzling StackExchange link. - Alexander Cooke Jul 16 2023

References

  • H. W. Gould, Combinatorial Identities, Morgantown, 1972, (1.77), page 10.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. also A006054, A006356, A005578, A075680.
Partial sums of A081294.
Cf. A002450, A004171, A007051, A083065, A083066, A083884.
Cf. A000978, A000979, A124400, A124401, A127936, A127955, A127956, A127957, A127958.
Cf. location of records in A007302.

Programs

  • GAP
    List([0..25], n-> (2^(2*n+1) + 1)/3); # G. C. Greubel, Dec 25 2019
  • Haskell
    a007583 = (`div` 3) . (+ 1) . a004171
-- Reinhard Zumkeller, Jan 09 2013
  • Magma
    [(2^(2*n+1) + 1)/3: n in [0..30] ]; // Vincenzo Librandi, Apr 28 2011
  • Maple
    a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-1 od: seq(a[n], n=0..23); # Zerinvary Lajos, Feb 22 2008, with correction by K. Spage, Aug 20 2014
A007583 := proc(n)
    (2^(2*n+1)+1)/3 ;
end proc: # R. J. Mathar, Feb 19 2015
  • Mathematica
    (* From Michael De Vlieger, Aug 22 2016 *)
Table[(2^(2n+1) + 1)/3, {n, 0, 23}]
Table[1 + 2Sum[4^k, {k, 0, n-1}], {n, 0, 23}]
NestList[4# -1 &, 1, 23]
Table[Sum[Binomial[n+k, 2k]/2^(k-n), {k, 0, n}], {n, 0, 23}]
CoefficientList[Series[(1-2x)/(1-5x+4x^2), {x, 0, 23}], x] (* End *)
  • PARI
    a(n)=sum(k=-n\3,n\3,binomial(2*n+1,n+1+3*k))
  • PARI
    a=1; for(n=1,23, print1(a,", "); a=bitor(a,3*a)) \\ K. Spage, Aug 20 2014
  • PARI
    Vec((1-2*x)/(1-5*x+4*x^2) + O(x^30)) \\ Altug Alkan, Dec 08 2015
  • PARI
    apply( {A007583(n)=2<<(2*n)\/3}, [0..25]) \\ M. F. Hasler, Nov 30 2021
  • Sage
    [(2^(2*n+1) + 1)/3 for n in (0..25)] # G. C. Greubel, Dec 25 2019

Formula

a(n) = 2*A002450(n) + 1.
From Wolfdieter Lang, Apr 24 2001: (Start)
a(n) = Sum_{m = 0..n} A060920(n, m) = A002450(n+1) - 2*A002450(n).
G.f.: (1-2*x)/(1-5*x+4*x^2). (End)
a(n) = Sum_{k = 0..n} binomial(n+k, 2*k)/2^(k - n).
a(n) = 4*a(n-1) - 1, n > 0.
From Paul Barry, Mar 17 2003: (Start)
a(n) = 1 + 2*Sum_{k = 0..n-1} 4^k;
a(n) = A001045(2n+1). (End)
a(n) = A020988(n-1) + 1 = A039301(n+1) - 1 = A083584(n-1) + 2. - Ralf Stephan, Jun 14 2003
a(0) = 1; a(n+1) = a(n) * 4 - 1. - Regis Decamps (decamps(AT)users.sf.net), Feb 04 2004 (correction to lead index by K. Spage, Aug 20 2014)
a(n) = Sum_{i + j + k = n; 0 <= i, j, k <= n} (n+k)!/i!/j!/(2*k)!. - Benoit Cloitre, Mar 25 2004
a(n) = 5*a(n-1) - 4*a(n-2). - Emeric Deutsch, Apr 01 2004
a(n) = 4^n - A001045(2*n). - Paul Barry, Apr 17 2004
a(n) = 2*(A001045(n))^2 + (A001045(n+1))^2. - Paul Barry, Jul 15 2004
a(n) = left and right terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A002450(n+1) a(n)] E.g. a(3) = 43 since M^n * [1 1 1] = [43 85 43] = [a(3) A002450(4) a(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = A072197(n) - A020988(n). - Creighton Dement, Dec 31 2004
a(n) = A139250(2^n). - Omar E. Pol, Feb 28 2011
a(n) = A193652(2*n+1). - Reinhard Zumkeller, Aug 08 2011
a(n) = Sum_{k = -floor(n/3)..floor(n/3)} binomial(2*n, n+3*k)/2. - Mircea Merca, Jan 28 2012
a(n) = 2^(2*(n+1)) - A072197(n). - Vladimir Pletser, Apr 12 2014
a(n) == 2*n + 1 (mod 3). Indeed, from Regis Decamps' formula (Feb 04 2004) we have a(i+1) - a(i) == -1 (mod 3), i= 0, 1, ..., n - 1. Summing, we have a(n) - 1 == -n (mod 3), and the formula follows. - Vladimir Shevelev, May 20 2015
For n > 0 a(n) = A133494(0) + 2 * (A133494(n) + Sum_{x = 1..n - 1}Sum_{k = 0..x - 1}(binomial(x - 1, k)*(A133494(k+1) + A133494(n-x+k)))). - J. Conrad, Dec 06 2015
a(n) = Sum_{k = 0..2n} (-2)^k == 1 + Sum_{k = 1..n} 2^(2k-1). - Bob Selcoe, Aug 21 2016
E.g.f.: (1 + 2*exp(3*x))*exp(x)/3. - Ilya Gutkovskiy, Aug 21 2016
A075680(a(n)) = 1, for n > 0. - Ralf Stephan, Jun 17 2025

A000978 Wagstaff numbers: numbers k such that (2^k + 1)/3 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

It is easy to see that the definition implies that k must be an odd prime. - N. J. A. Sloane, Oct 06 2006
The terms from a(32) on only give probable primes as of 2018. Caldwell lists the largest certified primes. - Jens Kruse Andersen, Jan 10 2018
Prime numbers of the form 1+Sum_{i=1..m} 2^(2i-1). - Artur Jasinski, Feb 09 2007
There is a new conjecture stating that a Wagstaff number is prime under the following condition (based on DiGraph cycles under the LLT): Let p be a prime integer > 3, N(p) = 2^p+1 and W(p) = N(p)/3, S(0) = 3/2 (or 1/4) and S(i+1) = S(i)^2 - 2 (mod N(p)). Then W(p) is prime iff S(p-1) == S(0) (mod W(p)). - Tony Reix, Sep 03 2007
As a member of the DUR team (Diepeveen, Underwood, Reix), and thanks to the LLR tool built by Jean Penne, I've found a new and big Wagstaff PRP: (2^4031399+1)/3 is Vrba-Reix PRP! This Wagstaff number has 1,213,572 digits and today is the 3rd biggest PRP ever found. I've done a second verification on a Nehalem core with the PFGW tool. - Tony Reix, Feb 20 2010
13347311 and 13372531 were found to be terms of this sequence (maybe not the next ones) by Ryan Propper in September 2013. - Max Alekseyev, Oct 07 2013
The next term is larger than 10 million. - Gord Palameta, Mar 22 2019
Ryan Propper found another likely term, 15135397, though it only corresponds to a probable prime. - Charles R Greathouse IV, Jul 01 2021

References

  • J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • S. S. Wagstaff, Jr., personal communication.

Links

Crossrefs

Cf. A107036 (indices of prime Jacobsthal numbers).
Cf. A000979, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A010051, A065091, A001045.

Programs

  • Haskell
    a000978 n = a000978_list !! (n-1)
a000978_list = filter ((== 1) . a010051 . a001045) a065091_list
-- Reinhard Zumkeller, Mar 24 2013
  • Mathematica
    Select[Range[5000], PrimeQ[(2^# + 1)/3] &] (* Michael De Vlieger, Jan 10 2018 *)
Select[Prime[Range[2,500]],PrimeQ[(2^#+1)/3]&] (* Harvey P. Dale, Jun 13 2022 *)
  • PARI
    forprime(p=2,5000,if(ispseudoprime(2^p\/3),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011
  • Python
    from gmpy2 import divexact
from sympy import prime, isprime
A000978 = [p for p in (prime(n) for n in range(2,10**2)) if isprime(divexact(2**p+1,3))] # Chai Wah Wu, Sep 04 2014

Formula

a(n) = A107036(n) for n>1. - Alexander Adamchuk, Feb 10 2007

Extensions

a(30) from Kamil Duszenko (kdusz(AT)wp.pl), Feb 03 2003; a(30) was proved prime by Francois Morain with FastECPP. - Tony Reix, Sep 03 2007
a(31)-a(39) from Robert G. Wilson v, Apr 11 2005
a(40) from Vincent Diepeveen (diep(AT)xs4all.nl) added by Alexander Adamchuk, Jun 19 2008
a(41) from Tony Reix, Feb 20 2010

A127936 Numbers k such that 1 + Sum_{i=1..k} 2^(2*i-1) is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 15, 21, 30, 39, 50, 63, 83, 95, 99, 156, 173, 350, 854, 1308, 1769, 2903, 5250, 5345, 5639, 6195, 7239, 21368, 41669, 47684, 58619, 63515, 69468, 70539, 133508, 134993, 187160, 493095
Offset: 1

Views

Author

Artur Jasinski, Feb 08 2007, Feb 09 2007

Keywords

Comments

If this sequence is infinite then so is A124401.
Equals A127965(n)/2.
The sum has the simple closed form 1 + 2/3*(4^n-1). - Stefan Steinerberger, Nov 24 2007
Terms beyond a(30) correspond to probable primes, cf. A000978. - M. F. Hasler, Aug 29 2008

Examples

			a(1)=1 because 1 + 2 = 3 is prime;
a(2)=2 because 1 + 2 + 2^3 = 11 is prime;
a(3)=3 because 1 + 2 + 2^3 + 2^5 = 43 is prime;
a(4)=5 because 1 + 2 + 2^3 + 2^5 + 2^7 + 2^9 = 683 is prime;
...

Crossrefs

Cf. A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936, A127936, A124401, A010051, A007583.

Programs

Formula

a(n) = floor(A000978(n)/2) = ceiling(log(4)(A000979(n))); A000978(n) = 2 a(n) + 1; A000979(n) = (2*4^a(n)+1)/3. - M. F. Hasler, Aug 29 2008

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 11 2007
2 more terms from Stefan Steinerberger, Nov 24 2007
6 more terms from Dmitry Kamenetsky, Jul 12 2008
a(30)-a(40) calculated from A000978 by M. F. Hasler, Aug 29 2008

A127962 Binary expansion of A000979(n).

Original entry on oeis.org

11, 1011, 101011, 1010101011, 101010101011, 1010101010101011, 101010101010101011, 1010101010101010101011, 101010101010101010101010101011, 101010101010101010101010101010101010101011
Offset: 1

Views

Author

Artur Jasinski, Feb 09 2007

Keywords

Links

Crossrefs

Cf. A000978, A000979, A007088, A124400, A126614, A127936, A127955, A127956, A127957, A127958, A127961.

Programs

  • Mathematica
    b[n_] := FromDigits[IntegerDigits[n, 2]]; b /@ Select[Table[(2^p + 1)/3, {p, Prime[Range[15]]}], PrimeQ] (* Amiram Eldar, Jul 23 2023 *)
  • Python
    from gmpy2 import divexact
from sympy import prime, isprime
A127962 = [int(bin(p)[2:]) for p in (divexact(2**prime(n)+1,3) for n in range(2,10**2)) if isprime(p)] # Chai Wah Wu, Sep 04 2014

Formula

a(n) = A007088(A000979(n)). - Amiram Eldar, Jul 23 2023

Extensions

Edited by N. J. A. Sloane, Jun 11 2007

A127963 Number of 1's in A127962(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 12, 16, 22, 31, 40, 51, 64, 84, 96, 100, 157, 174, 351, 855, 1309, 1770, 2904, 5251, 5346, 5640, 6196, 7240, 21369, 41670, 47685, 58620, 63516, 69469, 70540, 133509, 134994, 187161, 493096, 2015700
Offset: 1

Views

Author

Artur Jasinski, Feb 09 2007

Keywords

Crossrefs

Cf. A000120, A000978, A000979, A007953, A124400, A126614, A127936, A127955, A127956, A127957, A127958, A127961, A127962.

Programs

  • Mathematica
    b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 1]], {x, 1, Length[a]}]; d (* Artur Jasinski, Feb 09 2007 *)
DigitCount[#, 2, 1]& /@ Select[Table[(2^p + 1)/3, {p, Prime[Range[300]]}], PrimeQ] (* Amiram Eldar, Jul 23 2023 *)

Formula

a(n) = A000120(A000979(n)). - Michel Marcus, Nov 07 2013
a(n) = A007953(A127962(n)). - Amiram Eldar, Jul 23 2023

Extensions

a(22)-a(29) from Vincenzo Librandi, Mar 31 2012
a(30)-a(41) from Amiram Eldar, Jul 23 2023

A127964 Number of 0's in the binary expansion of A127962(n).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 14, 20, 29, 38, 49, 62, 82, 94, 98, 155, 172, 349, 853, 1307, 1768, 2902, 5249, 5344, 5638, 6194, 7238, 21367, 41668, 47683, 58618, 63514, 69467, 70538, 133507, 134992, 187159, 493094, 2015698
Offset: 1

Views

Author

Artur Jasinski, Feb 09 2007

Keywords

Comments

Apparently numbers k such that (2^(2*k+3)+1)/3 is prime. - James R. Buddenhagen, Apr 14 2011 [This is true. See the second formula. - Amiram Eldar, Oct 13 2024]

Crossrefs

Cf. A000978, A000979, A023416, A127962, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

Programs

  • Mathematica
    b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 0]], {x, 1, Length[a]}]; d
(Select[Prime[Range[200]], PrimeQ[(2^# + 1)/3] &] - 3)/2 (* Amiram Eldar, Oct 13 2024 *)

Formula

a(n) = A023416(A000979(n)). - Michel Marcus, Nov 07 2013
a(n) = (A000978(n)-3)/2. - Amiram Eldar, Oct 13 2024

Extensions

a(22)-a(29) from Vincenzo Librandi, Mar 31 2012
a(30)-a(41) from Amiram Eldar, Oct 13 2024

A127965 Number of bits in A127962(n).

Original entry on oeis.org

2, 4, 6, 10, 12, 16, 18, 22, 30, 42, 60, 78, 100, 126, 166, 190, 198, 312, 346, 700, 1708, 2616, 3538, 5806, 10500, 10690, 11278, 12390, 14478, 42736, 83338, 95368, 117238, 127030, 138936, 141078, 267016, 269986, 374320, 986190, 4031398
Offset: 1

Views

Author

Artur Jasinski, Feb 09 2007

Keywords

Crossrefs

Cf. A127962, A127963, A127964, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

Programs

  • Mathematica
    b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 0]+DigitCount[a[[x]], 10, 1]], {x, 1, Length[a]}]; d

Formula

a(n) = A127964(n) + A127963(n).
a(n) = 1 + floor(log_2(A000979(n))) = 1 + floor(log_2(2^A000978(n)+1) - A020857) = A000978(n) - 1. - R. J. Mathar, Feb 01 2008

Extensions

a(22)-a(29) from Vincenzo Librandi, Mar 30 2012
a(30)-a(41) from Amiram Eldar, Oct 19 2024

A194810 Indices k such that A139250(k) = A000979(n).

Original entry on oeis.org

2, 4, 8, 32, 64, 256, 512, 2048, 32768, 2097152, 1073741824, 549755813888, 1125899906842624, 9223372036854775808, 9671406556917033397649408, 39614081257132168796771975168, 633825300114114700748351602688
Offset: 1

Views

Author

Omar E. Pol, Oct 23 2011

Keywords

Comments

Indices k such that the number of toothpicks in the toothpick structure of A139250 after k-th stage equals the n-th Wagstaff prime A000979. All terms of this sequence are powers of 2 (see formulas).
For a picture of the n-th Wagstaff prime as a toothpick structure see the Applegate link "A139250: the movie version", then enter N = a(n) and click "Update", for N = a(n) <= 32768 (due to the resolution of the movie).

Examples

			For n = 5 we have that a(5) = 64, then we can see that the number of toothpicks in the toothpick structure of A139250 after 64 stages is 2731 which coincides with the fifth Wagstaff prime, so we can write A139250(64) = A000979(5) = 2731. See the illustration in the Applegate-Pol-Sloane paper, figure 3: T(64) = 2731 toothpicks.

Links

Crossrefs

Cf. A000978, A000979, A007583, A001045, A127936, A127958, A127962, A127965, A139250, A139253.

Programs

  • Mathematica
    2^Reap[Do[If[PrimeQ[1+Sum[2^(2n-1), {n, m}]], Sow[m]], {m, 100}]][[2, 1]] (* Jean-François Alcover, Oct 06 2018 *)

Formula

a(n) = 2^A127936(n) = 2^(floor(A000978(n)/2)) = 2^(ceiling(log_4(A000979(n)))).
A139250(2^n) = A007583(n), n >= 0.
A139250(a(n)) = A000979(n).

Extensions

More terms from Omar E. Pol, Mar 14 2012

A127959 Nonprime numbers of the form 1 + Sum_{k=1..m} 2^(2*k - 1).

Original entry on oeis.org

171, 10923, 699051, 11184811, 44739243, 178956971, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 11728124029611, 46912496118443, 187649984473771, 750599937895083, 3002399751580331, 12009599006321323
Offset: 1

Views

Author

Artur Jasinski, Feb 09 2007

Keywords

Comments

Prime numbers of the form 1 + Sum_{k=1..m} 2^(2*n - 1) is A000979. Numbers x such that 1 + Sum_{k=1..m} 2^(2*n - 1) is prime for n=1,2,...,x is A127936. A127955 is probably a subset of the present sequence.

Crossrefs

Cf. A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

Programs

  • Mathematica
    a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c] == False, AppendTo[a, c]], {x, 1, 50}]; a
Select[Table[Sum[2^(2k-1),{k,n}]+1,{n,50}],!PrimeQ[#]&] (* Harvey P. Dale, Dec 23 2017 *)
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