A139421 -id:A139421 - cp's OEIS frontend

A010693 Periodic sequence: Repeat 2,3.

2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 0

N. J. A. Sloane

a(n) = smallest prime divisor of n!! for n >= 2. For biggest prime divisor of n!! see A139421. - Artur Jasinski, Apr 21 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=-charpoly(A,-2). - Milan Janjic, Jan 27 2010
Simple continued fraction of 1+sqrt(5/3) = A176020. - R. J. Mathar, Mar 08 2012
p(n) = a(n-1) is the Abelian complexity function of the Thue-Morse word A010060. - Nathan Fox, Mar 12 2013

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 466
G. Richomme, K. Saari, and L. Q. Zamboni, Abelian complexity in minimal subshifts, J. London Math. Soc. 83(1) (2011) 79-95.
G. Richomme, K. Saari, and L. Q. Zamboni, Abelian complexity in minimal subshifts, arXiv:0911.2914 [math.CO], 2009.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A139421.

Cf. A026549 (partial products).

a010693 = (+ 2) . (`mod` 2)  -- Reinhard Zumkeller, Nov 27 2012
a010693_list = cycle [2,3]  -- Reinhard Zumkeller, Mar 29 2012

[2 + (n mod 2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2014

A010693:=n->2+(n mod 2): seq(A010693(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2014

Table[5/2 - (-1)^n/2, {n, 0, 100}] or a = {}; Do[b = First[First[FactorInteger[n!! ]]]; AppendTo[a, b], {n, 2, 1000}]; a (* Artur Jasinski, Apr 21 2008 *)
2 + Mod[Range[0, 100], 2] (* Wesley Ivan Hurt, Jul 24 2014 *)
PadRight[{},120,{2,3}] (* Harvey P. Dale, Jan 20 2023 *)

a(n)=3 - (n+1)%2 \\ Charles R Greathouse IV, May 09 2016

a(n) = 5/2 - ((-1)^n)/2.
a(n) = 2 + (n mod 2) = A007395(n) + A000035(n). - Reinhard Zumkeller, Mar 23 2005
a(n) = A020639(A016767(n)) for n>0. - Reinhard Zumkeller, Jan 29 2009
From Jaume Oliver Lafont, Mar 20 2009: (Start)
G.f.: (2+3*x)/(1-x^2).
Linear recurrence: a(0)=2, a(1)=3, a(n)=a(n-2) for n>=2. (End)
a(n) = A001615(2n)/A001615(n) for n > 0. - Enrique Pérez Herrero, Jun 06 2012
a(n) = floor((n+1)*5/2) - floor((n)*5/2). - Hailey R. Olafson, Jul 23 2014
a(n) = 3 - ((n+1) mod 2). - Wesley Ivan Hurt, Jul 24 2014
E.g.f.: 2*cosh(x) + 3*sinh(x). - Stefano Spezia, Aug 04 2025

Definition rewritten by Bruno Berselli, Sep 30 2011

A139426 Smallest number k such that M(n)^2+k*M(n)-1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

1, 5, 1, 5, 11, 11, 17, 19, 23, 97, 127, 145, 167, 269, 767, 479, 3307, 1453, 18007, 2357, 599, 17669, 5527, 3191, 3251, 70249, 147773, 39637

Offset: 1

Pierre CAMI, Apr 21 2008

All primes certified using openpfgw_v12 from primeform group.

			3*3+1*3-1=11 prime 3=M(1)=2^2-1 so k(1)=1;
7*7+5*7-1=83 prime 7=M(2)=2^3-1 so k(2)=5;
31*31+1*31-1=991 prime 31=M(3)=2^5-1 so k(3)=1.

Cf. A000668, A139424, A139425, A139427, A139428, A139429, A139430, A139421, A143385.

A000043 = {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};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
While[! PrimeQ[m2 + k*m - 1], k++]; k, {n, 15}] (* Robert Price, Apr 17 2019 *)

3 more terms. - Pierre CAMI, Aug 11 2008

A139424 Smallest number k such that M(n)^2-k*M(n)-1 is prime with M(n) = Mersenne primes = A000668(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 43, 1, 41, 53, 91, 317, 317, 43, 1, 37, 3595, 563, 17239, 911, 11497, 58501, 1259, 10283, 138569, 72247, 27733, 11777, 179105

Offset: 1

Pierre CAMI, Apr 21 2008

All primes certified using openpfgw_v12 from primeform group

			3*3-1*3-1=5 prime 3=M(1)=2^2-1 so k(1)=1;
7*7-1*7-1=41 prime 7=M(2)=2^3-1 so k(2)=1;
31*31-1*31-1=929 prime 31=M(3)=2^5-1 so k(3)=1.

Cf. A000668, A139425, A139426, A139427, A139428, A139429, A139430, A139421.

A000043 = {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};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
While[! PrimeQ[m2 - k*m - 1], k++]; k, {n, 15}] (* Robert Price, Apr 17 2019 *)

a(27)-a(28) from Robert Price, May 09 2019

A139425 Smallest number k such that M(n)^2-k*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

1, 1, 9, 3, 3, 25, 7, 21, 435, 241, 3, 153, 151, 493, 537, 2871, 1713, 4941, 4963, 307, 28413, 5035, 1615, 43525, 9973

Offset: 1

Pierre CAMI, Apr 21 2008

All primes certified using openpfgw_v12 from primeform group

			3*3-1*3+1=7 prime 3=M(1)=2^2-1 so k(1)=1;
7*7-1*7+1=43 prime 7=M(2)=2^3-1 so k(2)=1;
31*31-9*31+1=683 prime 31=M(3)=2^5-1 so k(3)=9.

Cf. A000668, A139424, A139426, A139427, A139428, A139429, A139430, A139421.

A000043 = {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};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
While[! PrimeQ[m2 - k*m + 1], k++]; k, {n, 15}] (* Robert Price, Apr 17 2019 *)

A139427 Smallest number k such that M(n)^2+k*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

1, 3, 5, 17, 17, 5, 83, 63, 71, 101, 543, 59, 569, 1029, 353, 1851, 2801, 2619, 525, 2907, 8955, 437, 30159, 5409, 8355

Offset: 1

Pierre CAMI, Apr 21 2008

All primes certified using openpfgw_v12 from primeform group

			3*3+1*3+1=13 prime 3=M(1)=2^2-1 so k(1)=1;
7*7+3*7+1=71 prime 7=M(2)=2^3-1 so k(2)=3;
31*31+5*31+1=1117 prime 31=M(3)=2^5-1 so k(3)=5.

Cf. A000668, A139424, A139425, A139426, A139428, A139429, A139430, A139421.

A000043 = {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};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; k = 1;
While[! PrimeQ[m2 + k*m + 1], k++]; k, {n, 15}] (* Robert Price, Apr 17 2019 *)

A139428 Smallest prime p such that M(n)^2-p*M(n)-1 is prime with M(n)= Mersenne primes =A000668(n).

Original entry on oeis.org

5, 7, 5, 17, 43, 67, 41, 53, 311, 317, 317, 43, 1427, 37, 25693, 563, 17239, 911, 11497, 112247, 1259, 190639, 138569, 296713, 27733, 11777

Offset: 2

Pierre CAMI, Apr 21 2008

All primes certified using openpfgw_v12 from primeform group

			7*7-5*7-1=13 prime 7=M(2)=2^3-1 so k(2)=5;
31*31-7*31-1=743 prime 31=M(3)=2^5-1 so k(3)=7.

Cf. A000668, A139424, A139425, A139426, A139427, A139429, A139430, A139421.

A000043 = {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};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; p = 1;
While[! PrimeQ[m2 - Prime[p]*m - 1], p++];
Prime[p], {n, 15}] (* Robert Price, Apr 17 2019 *)

A139429 Smallest prime p such that M(n)^2 - p*M(n) + 1 is prime with M(n) = A000668(n).

Original entry on oeis.org

3, 19, 3, 3, 73, 7, 271, 1021, 241, 3, 487, 151, 2971, 35839, 5737, 1723, 81943, 115741, 307, 151549, 231823, 443431, 195163, 9973, 114913, 362599

Offset: 2

Pierre CAMI, Apr 21 2008

All primes certified using openpfgw_v12 from primeform group.

			7*7-3*7+1=29 prime 7=M(2)=2^3-1 so k(2)=3;
31*31-19*31+1=373 prime 31=M(3)=2^5-1 so k(3)=19.

Cf. A000668, A139424, A139425, A139426, A139427, A139428, A139430, A139421.

A000043 = {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};
Table[m = 2^A000043[[n]] - 1; m2 = m^2; p = 1;
While[! PrimeQ[m2 - Prime[p]*m + 1], p++];
Prime[p], {n, 15}] (* Robert Price, Apr 17 2019 *)

cp's OEIS Frontend

A010693 Periodic sequence: Repeat 2,3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A139426 Smallest number k such that M(n)^2+k*M(n)-1 is prime with M(n)= Mersenne primes =A000668(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A139424 Smallest number k such that M(n)^2-k*M(n)-1 is prime with M(n) = Mersenne primes = A000668(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A139425 Smallest number k such that M(n)^2-k*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A139427 Smallest number k such that M(n)^2+k*M(n)+1 is prime with M(n)= Mersenne primes =A000668(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A139428 Smallest prime p such that M(n)^2-p*M(n)-1 is prime with M(n)= Mersenne primes =A000668(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A139429 Smallest prime p such that M(n)^2 - p*M(n) + 1 is prime with M(n) = A000668(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica