A069362 -id:A069362 - cp's OEIS frontend

A359576 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with a path of adjacent 1's from top row to bottom row.

1, 3, 1, 7, 7, 1, 15, 37, 17, 1, 31, 175, 197, 41, 1, 63, 781, 1985, 1041, 99, 1, 127, 3367, 18621, 22193, 5503, 239, 1, 255, 14197, 167337, 433809, 247759, 29089, 577, 1, 511, 58975, 1461797, 8057905, 10056959, 2764991, 153769, 1393, 1, 1023, 242461, 12519345, 144769425, 384479935, 232824241, 30856705, 812849, 3363, 1

Offset: 1

Andrew Howroyd, Jan 06 2023

The grid has m rows and n columns.
"Path" refers to a sequence of L(eft), R(ight), U(p), D(own) steps (edge connectivity like in fixed polyominoes), self-avoiding, starting anywhere in the first row and ending anywhere in the last row. The path does not need to step on all 1's of the array. The path has obviously at least m-1 steps. - R. J. Mathar, Jun 21 2023
Note that the total would be smaller if Up steps were disallowed (as in the original comment above); the smallest grid size for which this phenomenon occurs is 4 X 5. The total number of 4 X 5 and 5 X 5 grids would be 433801 instead of 433809 and 10056087 instead of 10056959, respectively, without Up steps. - Caleb Stanford, Feb 01 2024
Each row and each column satisfies a linear recurrence with constant coefficients. - Pontus von Brömssen, Feb 05 2025

			Array begins:
====================================================================
m\n| 1   2      3        4          5            6             7
---+----------------------------------------------------------------
1  | 1   3      7       15         31           63           127 ...
2  | 1   7     37      175        781         3367         14197 ...
3  | 1  17    197     1985      18621       167337       1461797 ...
4  | 1  41   1041    22193     433809      8057905     144769425 ...
5  | 1  99   5503   247759   10056959    384479935   14142942975 ...
6  | 1 239  29089  2764991  232824241  18287614751 1374273318721 ...
7  | 1 577 153769 30856705 5388274121 868972410929 ...
  ...
All the 37 2 X 3 binary arrays:
001 001 001 001
001 011 101 111 plus 4 copies left-right flipped
.
010 010 010 010
010 011 110 111
.
011 011 011 011 011 011
001 010 011 101 110 111 plus 6 copies left-right flipped
.
101 101 101 101 101 101
001 011 100 101 110 111
.
111 111 111 111 111 111 111
001 010 011 100 101 110 111 - _R. J. Mathar_, Jun 21 2023

Samuel Dittmer, Hiram Golze, Grant Molnar, and Caleb Stanford, Puzzle and Proof: A Decade of Problems from the Utah Math Olympiad, CRC Press, 2025, p. 51.

Caleb Stanford, Rust program to compute the sequence.
Utah Math Olympiad 2022, Problem 6.

Main diagonal is A365988.
Rows 1..19 are A000225, A005061, A069361, A069362, A069363-A069377.
Columns 1..20 are A000012, A001333(n+1), A069378, A069379, A069380-A069395.
Cf. A359574, A359575.

One additional diagonal of terms added by Caleb Stanford, Feb 05 2024

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

Original entry on oeis.org

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

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

A368809 Number of 4 X n binary arrays with a path of adjacent 1's from top row to bottom row using only left, right, and downward steps.

Original entry on oeis.org

1, 41, 1041, 22193, 433801, 8057625, 144762849, 2540882465, 43840779353, 746649798473, 12587443678705, 210491254232465, 3496816762316713, 57778098654714361, 950391251581267073, 15574198350636963201, 254405750326548970361, 4144508602760970898729, 67361936661916258817937

Offset: 1

Caleb Stanford, Feb 05 2024

Unlike A069362, does not allow upward steps.

			For example, here is one such 4 X 4 array:
    0001
    1111
    1010
    1100
The following 4 X 5 array is a non-example, as there is no path using only left, right, and downward steps:
    10000
    10111
    11101
    00001

Index entries for linear recurrences with constant coefficients, signature (36,-413,1642,-2536,1152).

Row 4 of A369892.

Cf. A069362, A359576.

G.f.: x*(1 + 5*x - 22*x^2 + 8*x^3)/((1 - 16*x)*(1 - 20*x + 93*x^2 - 154*x^3 + 72*x^4)). - Pontus von Brömssen, Feb 05 2025

More terms from Pontus von Brömssen, Feb 05 2025

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

A359576 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with a path of adjacent 1's from top row to bottom row.

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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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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.