A059760 -id:A059760 - cp's OEIS frontend

A059763 Primes starting a Cunningham chain of the first kind of length 4.

509, 1229, 1409, 2699, 3539, 6449, 10589, 11549, 11909, 12119, 17159, 19709, 19889, 22349, 26189, 27479, 30389, 43649, 55229, 57839, 60149, 71399, 74699, 75329, 82499, 87539, 98369, 101399, 104369, 112919, 122099, 139439, 148829, 166739

Offset: 1

Labos Elemer, Feb 20 2001

Initial (unsafe) primes of Cunningham chains of first type with length exactly 4. Primes in A059453 that survive as primes just three "2p+1 iterations", forming chains of exactly 4 terms.
The definition indicates each chain is exactly 4 primes long (i.e., the chain cannot be a subchain of a longer one). That is why this sequence is different from A023272, which also gives primes included in longer chains ("starting" them or not).
Prime p such that {(p-1)/2, p, 2p+1, 4p+3, 8p+7, 16p+15} = {composite, prime, prime, prime, prime, composite}.

			1229 is a term because, through 2p+1, 1229 -> 2459 -> 4919 -> 9839 and the chain ends here since 2*9839 + 1 = 11*1789 is composite.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700.

Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

isA059763 := proc(p) local pitr,itr ; if isprime(p) then if isprime( (p-1)/2 ) then RETURN(false) ; else pitr := p ; for itr from 1 to 3 do pitr := 2*pitr+1 ; if not isprime(pitr) then RETURN(false) ; fi ; od: pitr := 2*pitr+1 ; if isprime(pitr) then RETURN(false) ; else RETURN(true) ; fi ; fi ; else RETURN(false) ; fi ; end: for i from 2 to 100000 do p := ithprime(i) ; if isA059763(p) then printf("%d,",p) ; fi ; od: # R. J. Mathar, Jul 23 2008

Edited and extended by R. J. Mathar, Jul 23 2008, Aug 18 2008

A059764 Initial (unsafe) primes of Cunningham chains of first type with length exactly 5. Primes in A059453 that survive as primes just four "2p+1 iterations", forming chains of exactly 5 terms.

Original entry on oeis.org

2, 53639, 53849, 61409, 66749, 143609, 167729, 186149, 206369, 268049, 296099, 340919, 422069, 446609, 539009, 594449, 607319, 658349, 671249, 725009, 775949, 812849, 819509, 926669, 1008209, 1092089, 1132949, 1271849

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3, 8p+7, 16p+15, 32p+31} = {nonprime, prime, prime, prime, prime, prime, composite}.

			2 is here because (2-1)/2 = 1/2 and 32*2+31 = 95 are not primes, while 2, 5, 11, 23, and 47 give a first-kind Cunningham chain of 5 primes which cannot be continued.
53639 is here because through <2p+1>, 53639 -> 107279 -> 214559 -> 429119 -> 858239 and the chain ends here (with this operator).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452-A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

l5Q[n_]:=Module[{a=PrimeQ[(n-1)/2],b=PrimeQ[ NestList[2#+1&,n,5]]}, Join[{a},b]=={False,True,True,True,True,True,False}]; Select[Range[ 1300000],l5Q] (* Harvey P. Dale, Oct 14 2012 *)

Definition corrected by Alexandre Wajnberg, Aug 31 2005

Entry revised by N. J. A. Sloane, Apr 01 2006

A006231 a(n) = Sum_{k=2..n} n(n-1)...(n-k+1)/k.

Original entry on oeis.org

0, 1, 5, 20, 84, 409, 2365, 16064, 125664, 1112073, 10976173, 119481284, 1421542628, 18348340113, 255323504917, 3809950976992, 60683990530208, 1027542662934897, 18430998766219317, 349096664728623316, 6962409983976703316, 145841989688186383337

Offset: 1

R. K. Guy

a(n) is also the number of permutations in the symmetric group S_n that are pure cycles, see example. - Avi Peretz (njk(AT)netvision.net.il), Mar 24 2001
Also the number of elementary circuits in a complete directed graph with n nodes [D. B. Johnson, 1975]. - N. J. A. Sloane, Mar 24 2014
If one takes 1,2,3,4, ..., n and starts creating parenthetic products of k-tuples and adding, one gets a(n+1). For 1,2,3,4 one gets (1)+(2)+(3)+(4) = 10; (1*2)+(2*3)+(3*4) = 20; (1*2*3)+(2*3*4) = 30; (1*2*3*4) = 24; and 10+20+30+24 = 84 = a(5). - J. M. Bergot, Apr 24 2014
Let P_n be the set of probability distributions over orderings of n objects that can be obtained by drawing n real numbers from independent probability distributions and sorting. Then a(n) is conjectured to be the dimension of P_n, as a semi-algebraic subset of R^(n!). - Jamie Tucker-Foltz, Jul 29 2024

			a(3) = 5 because the cycles in S_3 are (12), (13), (23), (123), (132).
a(4) = 20 because there are 24 permutations of {1,2,3,4} but we don't count (12)(34), (13)(24), (14)(23) or the identity permutation. - _Geoffrey Critzer_, Nov 03 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from T. D. Noe)
Eric Babson, Moon Duchin, Annina Iseli, Pietro Poggi-Corradini, Dylan Thurston, Jamie Tucker-Foltz, Models of Random Spanning Trees, arXiv:2407.20226 [math.CO], 2023.
R. K. Guy, Letter to N. J. A. Sloane, 1977
Donald B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Comput. 4 (1975), 77-84. MR0398155 (53 #2010).
Jamie Tucker-Foltz, Code to compute dimension of P_n on GitHub.

Cf. A059760, A000295.

Column k=1 of A136394.

a006231 n = numerator $
   sum $ tail $ zipWith (%) (scanl1 (*) [n,(n-1)..1]) [1..n]
-- Reinhard Zumkeller, Dec 27 2011

A006231 := proc(n)
    n*( hypergeom([1,1,1-n],[2],-1)-1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Aug 06 2013

a[n_] = n*(HypergeometricPFQ[{1,1,1-n}, {2}, -1] - 1); Table[a[n], {n, 1, 20}] (* Jean-François Alcover,  Mar 29 2011 *)
Table[Sum[Times@@Range[n-k+1,n]/k,{k,2,n}],{n,20}] (* Harvey P. Dale, Sep 23 2011 *)

a(n) = n--; sum(ip=1, n, sum(j=1, n-ip+1, prod(k=j, j+ip-1, k))); \\ Michel Marcus, May 07 2014 after comment by J. M. Bergot

a(n+1) - a(n) = A000522(n) - 1.
a(n) = n*( 3F1(1,1,1-n; 2;-1) -1). - Jean-François Alcover,  Mar 29 2011
E.g.f.: exp(x)*(log(1/(1-x))-x). - Geoffrey Critzer, Sep 12 2012
G.f.: (Q(0) - 1)/(1-x)^2, where Q(k)= 1 + (2*k + 1)*x/( 1 - x - 2*x*(1-x)*(k+1)/(2*x*(k+1) + (1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 09 2013
Conjecture: a(n) + (-n-2)*a(n-1) + (3*n-2)*a(n-2) + 3*(-n+2)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Aug 06 2013

More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001

A059766 Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.

Original entry on oeis.org

89, 63419, 127139, 405269, 810809, 1069199, 1178609, 1333889, 1598699, 1806089, 1958249, 2606069, 2848949, 3241289, 3339989, 3784199, 3962039, 4088879, 4444829, 4664249, 4894889, 4897709, 5132999, 5215499, 5238179, 6026309, 6059519, 6088529, 6490769, 6676259

Offset: 1

Labos Elemer, Feb 21 2001

nonn

Special terms of A059453. Not identical to A023330 of which 1122659, 2164229, 2329469, ..., etc. are omitted since they have exact length 7 or larger.

Unsafe primes starting complete chains of length 6.

			89 is a term because (89-1)/2 = 44 and 64*89+63 = 5759 = 13*443 are composites, while 89, 179, 359, 719, 1439, and 2879 are primes.
1122659 is not a term because it initiates a chain of length 7.
4658939 is not a term because (4658939-1)/2 = 2329469 is prime. - _Sean A. Irvine_, Oct 09 2022

Amiram Eldar, Table of n, a(n) for n = 1..1000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Cf. A005602, A053176, A059500, A057331, A059688, A059690.

Entry revised by N. J. A. Sloane Apr 01 2006

a(12) onward corrected and extended by Sean A. Irvine, Oct 09 2022

A110024 Smallest primes starting a complete three iterations Cunningham chain of the second kind.

Original entry on oeis.org

2131, 2311, 6211, 7411, 10321, 18121, 22531, 23011, 24391, 29671, 31771, 35311, 41491, 46411, 54601, 56311, 60331, 61381, 67651, 78031, 85381, 96931, 99871, 109471, 126001, 134731, 156691, 162451, 165331, 170851, 185131, 205171, 224401

Offset: 1

Alexandre Wajnberg, Sep 03 2005

The word "complete" indicates each chain is exactly 4 primes long (i.e., the chain cannot be a subchain of another one). Other sequences give also primes included in longer chains ("starting" them or not).

Terms computed by Gilles Sadowski.

			2311 is here because, through the operator <*2-1> of the chains of the second kind,
2311 -> 4621 -> 9241 -> 18481 and the chain ends here (with this operator).

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700

Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Edited and extended by R. J. Mathar, May 08 2009

A110022 Primes starting a Cunningham chain of the second kind of length 5.

Original entry on oeis.org

1531, 6841, 15391, 44371, 57991, 83431, 105871, 145021, 150151, 199621, 209431, 212851, 231241, 242551, 291271, 319681, 346141, 377491, 381631, 451411, 481021, 506791, 507781, 512821, 537811, 588871, 680431, 727561, 749761, 782911, 787711

Offset: 1

Alexandre Wajnberg, Sep 03 2005

The definition indicates that each chain is exactly 5 primes long (i.e. the chain cannot be a subchain of a longer one). That's why this sequence is different from A057328 which gives also primes included in longer chains (thus not "starting" them), as 16651, starting a seven primes chain, or 33301, second prime of the same seven primes chain.

			6841 is here because: 6841 through <2p-1> -> 13681-> 27361-> 54721-> 109441 and the chain ends here since 2*109441-1=13*113*149 is composite.

Chris Caldwell's Prime Glossary, Cunningham chains.
G. Löh, Long chains of nearly doubled primes, Math. Comp. vol. 53 no. 188 (1989) pp 751-759.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

isA110022 := proc(p) local pitr,itr ; if isprime(p) then if isprime( (p+1)/2 ) then RETURN(false) ; else pitr := p ; for itr from 1 to 4 do pitr := 2*pitr-1 ; if not isprime(pitr) then RETURN(false) ; fi ; od: pitr := 2*pitr-1 ; if isprime(pitr) then RETURN(false) ; else RETURN(true) ; fi ; fi ; else RETURN(false) ; fi ; end: for i from 2 to 200000 do p := ithprime(i) ; if isA110022(p) then printf("%d,",p) ; fi ; od: # R. J. Mathar, Jul 23 2008

Edited and extended by R. J. Mathar, Jul 23 2008

A110025 Smallest primes starting a complete three iterations Cunningham chain of the first or second kind.

Original entry on oeis.org

509, 1229, 1409, 2131, 2311, 2699, 3539, 6211, 6449, 7411, 10321, 10589, 11549, 11909, 12119, 17159, 18121, 19709, 19889, 22349, 22531, 23011, 24391, 26189, 27479, 29671, 30389, 31771, 35311, 41491, 43649, 46411, 54601, 55229, 56311

Offset: 1

Alexandre Wajnberg, Sep 03 2005

Terms computed by Gilles Sadowski.

			1409 is here because, through the operator <2p+1> for chains of the first kind, 1409 -> 2819 -> 5639 -> 11279 and the chain ends here.
2131 is here because, through the operator <2p-1> for chains of the second kind, 2131 -> 4261 -> 8521 -> 17041 and the chain ends here.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700.

Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Union of A059763 and A110024. [R. J. Mathar, May 08 2009]

Edited by R. J. Mathar, May 08 2009

A110027 Smallest primes starting a complete four iterations Cunningham chain of the first or second kind.

Original entry on oeis.org

2, 1531, 6841, 15391, 44371, 53639, 53849, 57991, 61409, 66749, 83431, 105871, 143609, 145021, 150151, 167729, 186149, 199621, 206369, 209431, 212851, 231241, 242551, 268049, 291271, 296099, 319681, 340919, 346141, 377491, 381631, 422069

Offset: 1

Alexandre Wajnberg, Sep 03 2005

The word "complete" indicates each chain is exactly 5 primes long (i.e., the chain cannot be a subchain of another one).

Terms computed by Gilles Sadowski.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700,

Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

Union of A059764 and A110022 . [R. J. Mathar, May 08 2009]

Edited and extended by R. J. Mathar, May 08 2009

A059615 a(n) is the number of non-parallel lines determined by a pair of vertices (extreme points) in the polytope of real n X n doubly stochastic matrices. The vertices are the n! permutation matrices.

Original entry on oeis.org

0, 1, 15, 240, 6040, 217365, 10651011, 681667840, 55215038880, 5521504648185, 668102052847735, 96206695728917136, 16258931576714668920, 3186750589054271109325, 717018882536990087693835

Offset: 1

Noam Katz (noamkj(AT)hotmail.com), Feb 18 2001

nonn

			a(3) = 15 because there are 3! = 6 vertices and C(6,2) lines and in this case all are nonparallel so a(3) = C(6,2) = 15.

M. Marcus, Hermitian Forms and Eigenvalues, in Survey of Numerical Analysis, J. Todd, ed. McGraw-Hill, New York, 1962.

Cf. A000166, A059760.

Digits := 200: with(combinat): d := n->n!*sum((-1)^j/j!,j=0..n): a059615 := n->1/2*sum( binomial(n,k)^2 * (n-k)!*d(n-k), k=0..n-2): for n from 1 to 30 do printf(`%d,`,round(evalf(a059615(n)))) od:

a(n) = (1/2)*sum(k=0, n-2, ((n!/k!)^2 * sum(m=0, n-k, (-1)^m/m!))); \\ Michel Marcus, Mar 14 2018

a(n) = (1/2)*Sum_{k=0...n-2} binomial(n,k)^2 * (n-k)! * d(n-k) for n >= 2, where d(n) is the number of derangements of n elements: permutations of n elements with no fixed points - sequence A000166. Using the formula: d(n) = n!*Sum_{k=0..n} (-1)^k/k!, a(n) = (1/2)*Sum_{k=0..n-2} ((n!/k!)^2 * Sum_{m=0..n-k} (-1)^m/m!).

More terms from James Sellers, Feb 19 2001

Offset corrected by Michel Marcus, Mar 14 2018

A110089 Smallest prime beginning (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

11, 3, 2, 509, 2, 89, 16651, 15514861, 85864769, 26089808579, 665043081119, 554688278429, 758083947856951, 95405042230542329, 69257563144280941

Offset: 1

Alexandre Wajnberg, Sep 04 2005

The word "complete" indicates each chain is exactly n primes long for the operator in function (i.e. the chain cannot be a subchain of another one); and the first and/or last term may be involved in a chain of the other kind (i.e. the chain may be connected to another one). a(1)-a(8) computed by Gilles Sadowski.

			a(1)=11 because 2, 3, 5 and 7 are included in longer chains than one prime long; and 11 (although included in a <2p+1> chain) has no prime connection through <2p-1>.
a(2)=3 because 3 begins (through 2p+1>) the first complete two primes chain: 3-> 7 (even if 3 and 7 are also part of two others chains, but through <2p-1>).
a(3)=2 because (although 2 begins also a five primes chain through <2p+1>) it begins, through <2p-1>, the first complete three primes chain encountered: 2->3->5.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

a(n) = min(A005602(n), A005603(n)). - R. J. Mathar, Jul 23 2008

a(8)-a(13) via A005602, A005603 from R. J. Mathar, Jul 23 2008

a(14)-a(15) via A005602, A005603 from Jason Yuen, Sep 03 2024

cp's OEIS Frontend

A059763 Primes starting a Cunningham chain of the first kind of length 4.

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A059764 Initial (unsafe) primes of Cunningham chains of first type with length exactly 5. Primes in A059453 that survive as primes just four "2p+1 iterations", forming chains of exactly 5 terms.

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A006231 a(n) = Sum_{k=2..n} n(n-1)...(n-k+1)/k.

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A059766 Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.

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A110024 Smallest primes starting a complete three iterations Cunningham chain of the second kind.

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A110022 Primes starting a Cunningham chain of the second kind of length 5.

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A110025 Smallest primes starting a complete three iterations Cunningham chain of the first or second kind.

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A110027 Smallest primes starting a complete four iterations Cunningham chain of the first or second kind.

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A110089 Smallest prime beginning (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.