A007700 -id:A007700 - cp's OEIS frontend

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

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

A059688 Length of Cunningham chain containing prime(n) either as initial, internal or final term.

Original entry on oeis.org

5, 2, 5, 2, 5, 0, 0, 0, 5, 2, 0, 0, 3, 0, 5, 2, 2, 0, 0, 0, 0, 0, 3, 6, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 2, 6, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Labos Elemer, Feb 06 2001

nonn

The length of a chain is measured by the total number of terms including the end points. a(n)=0 means that prime(n) is neither Sophie Germain nor a safe prime (i.e. it is in A059500).

			For all of {2,5,11,23,47}, i.e. at positions {j}={1,3,5,9,15} a(j)=5. Similarly for indices of all terms in {89,...,5759} a(i)=6. No chains are intelligible with length = 1 because the minimal chain enclose one Sophie Germain and also one safe prime. Dominant values are 0 and 2.

C. K. Caldwell, Cunningham Chains
W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A059500.

Offset and a(5) corrected by Sean A. Irvine, Oct 01 2022

A059767 Initial (unsafe) primes of Cunningham chains of first type with length exactly 7.

Original entry on oeis.org

1122659, 2164229, 2329469, 10257809, 10309889, 12314699, 14030309, 14145539, 23103659, 24176129, 28843649, 37088729, 42389519, 49160099, 50785439, 62800169, 68718059, 88174049, 95831189, 105388169, 121255889, 138140729, 155439419, 159938459, 173285999

Offset: 1

Labos Elemer, Feb 21 2001

nonn

Special primes from A059453.

Primes p such that (2^k)*p+(2^k)-1 is also prime for k = 0, 1, 2, 3, 4, 5, 6 and is composite for k = -1 and k = 7.

			C7 prime chain is generated from prime a(10) = 24176129 with 2p+1 iterations: 24176129, 48352259, 96704519, 193409039, 386818079, 773636159, 1547272319, 3094544639.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 178 (Rev. ed. 1997).

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A005602, A007700, A023272, A023302, A023330, A059452, A059453, A059454, A059455.

Transpose[Select[{#, Length[NestWhileList[2#+1&, #, PrimeQ]]-1}&/@ Prime[Range[PrimePi[24177000]]], #[[2]]>6&]][[1]]
Select[Prime[Range[10^6]], PrimeQ[a1=2*#+1]&&PrimeQ[a2=2*a1+1]&&PrimeQ[a3=2*a2+1]&&PrimeQ[a4=2*a3+1]&&PrimeQ[a5=2*a4+1]&&PrimeQ[a6=2*a5+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

is(n)=n%30==29 && isprime(n) && isprime(2*n+1) && isprime(4*n+3) && isprime(8*n+7) && isprime(16*n+15) && isprime(32*n+31) && isprime(64*n+63) && !isprime(n\2) && !isprime(128*n+127) \\ Charles R Greathouse IV, Dec 01 2016

Corrected and extended by Harvey P. Dale, Jul 10 2002
More terms from Vladimir Joseph Stephan Orlovsky, Jan 17 2009
Corrected by John Cerkan, Nov 30 2016

A162019 Double-safe primes which are also double-Sophie Germain primes.

Original entry on oeis.org

11, 359, 719, 214559, 215399, 245639, 253679, 266999, 507359, 508559, 574439, 670919, 744599, 825479, 1017119, 1072199, 1184399, 1363679, 1621079, 1688279, 1786439, 2156039, 2377799, 2429279, 2633399, 2684999, 2900039, 3103799

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 24 2009

nonn

The intersection of the primes in A066179 and those in A007700: they remain prime after each of two successive applications of the substitution p->(p-1)/2, and remain prime after each two successive applications of the substitution p->2p+1.

			a(1)=11 is double safe: (11-1)/2=5; (5-1)/2=2, and double Sophie-Germain: 2*11+1=23; 2*23+1=47.

Cf. A007700, A023302, A066179, A157359.

lst={};Do[p=Prime[n];If[PrimeQ[safe=(p-1)/2],If[PrimeQ[(safe-1)/2],If[PrimeQ[sophie=2*p+1],If[PrimeQ[2*sophie+1],AppendTo[lst,p]]]]],{n,3*9!}];lst

a(n) = 4*A023302(n) + 3 = (A157359(n)-3)/4. - R. J. Mathar, Jun 26 2009

Edited by R. J. Mathar, Jun 26 2009

A059690 Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.

Original entry on oeis.org

1, 2, 2, 2, 3, 5, 7, 13, 20, 31, 52, 83, 142, 242, 412, 742, 1308, 2294, 4040, 7327, 13253, 24255, 44306, 81700, 150401, 277335, 513705, 954847, 1780466, 3325109, 6224282, 11676337, 21947583, 41327438

Offset: 1

Labos Elemer, Feb 06 2001

			a(11)-a(10) = 21 means that between 1024 and 2048 exactly 21 primes introduce Cunningham chains: {1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003}.
Their lengths are 2, 3 or 4. Thus the complete chains spread over more than one binary size-zone: {1409, 2819, 5639, 11279}. The primes 1439 and 2879 also form a chain but 1439 is not at the beginning of that chain, 89 is.

C. K. Caldwell, Cunningham Chains
W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains

Cf. A023272, A023302, A023330, A005602, A007700, A053176, A059452-A059456, A059500, A057331, A059688, A007053, A036378, A029837, A007053.

c = 0; k = 1; Do[ While[k <= 2^n, If[ PrimeQ[k] && !PrimeQ[(k - 1)/2] && PrimeQ[2k + 1], c++ ]; k++ ]; Print[c], {n, 1, 29}]

from itertools import count, islice
from sympy import isprime, primerange
def c(p): return not isprime((p-1)//2) and isprime(2*p+1)
def agen():
    s = 1
    for n in count(2):
        yield s; s += sum(1 for p in primerange(2**(n-1)+1, 2**n) if c(p))
print(list(islice(agen(), 20))) # Michael S. Branicky, Oct 09 2022

Edited and extended by Robert G. Wilson v, Nov 23 2002
Title and a(30)-a(31) corrected, and a(32) from Sean A. Irvine, Oct 02 2022
a(33)-a(34) from Michael S. Branicky, Oct 09 2022

A122173 Expansion of -x * (x^5+x^4-15x^3+19x^2-8x+1) / (x^6-12x^5+34x^4-30x^3+6x^2+3x-1).

Original entry on oeis.org

1, -5, 10, -45, 110, -421, 1148, -4037, 11697, -39250, 117736, -384657, 1177235, -3787218, 11727187, -37389217, 116571621, -369712938, 1157315631, -3659226205, 11481436216, -36237006073, 113856243558, -358967583724, 1128781753801, -3556642214960, 11189229179710

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
Index entries for linear recurrences with constant coefficients, signature (3,6,-30,34,-12,1).

Cf. A046854. Cf. A046854. Cf. A007700, A059455. Cf. A065941.

M = {{0, -1, -1, -1, -1, -1}, {-1, 0, -1, -1, -1, 0}, {-1, -1, 0, -1, 0, 0}, {-1, -1, -1, 1, 0, 0}, {-1, -1, 0, 0, 1, 0}, {-1, 0, 0, 0, 0, 1}}; v[1] = {1, 1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Floor[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{3,6,-30,34,-12,1},{1,-5,10,-45,110,-421},30] (* Harvey P. Dale, Mar 16 2025 *)

G.f.: -x*(x^5+x^4-15*x^3+19*x^2-8*x+1)/(x^6-12*x^5+34*x^4-30*x^3+6*x^2+3*x-1). [Colin Barker, Oct 19 2012]

Sequence edited by Joerg Arndt, Colin Barker, Bruno Berselli, Oct 19 2012

A122174 First row sum of the matrix M^n, where M is the 5 X 5 matrix {{0,-1,-1,-1,-1}, {-1,0,-1,-1,0}, {-1,-1,0,0,0}, {-1,-1,0,1,0}, {-1,0,0,0,1}}.

Original entry on oeis.org

1, -4, 6, -24, 41, -145, 273, -886, 1789, -5457, 11605, -33807, 74761, -210366, 479256, -1313465, 3061242, -8222492, 19501429, -51579259, 123983182, -324067194, 787044384, -2038584810, 4990387355, -12836179872, 31614557443, -80883958143, 200146505560, -509959672813

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
Index entries for linear recurrences with constant coefficients, signature (2,5,-13,7,-1).

Cf. A046854, A007700, A059455, A065941.

with(linalg): M[1]:=matrix(5,5,[0,-1,-1,-1,-1,-1,0,-1,-1,0,-1,-1,0,0,0,-1,-1,0,1, 0,-1,0,0,0,1]): for n from 2 to 30 do M[n]:=multiply(M[n-1],M[1]) od: 1,seq(M[n][1,1]+M[n][1,2]+M[n][1,3]+M[n][1,4]+M[n][1,5],n=1..30);

M = {{0, -1, -1, -1, -1}, {-1, 0, -1, -1, 0}, {-1, -1, 0, 0, 0}, {-1, -1, 0, 1, 0}, {-1, 0, 0, 0, 1}}; v[1] = {1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 25}]

a(n) = my(m=[0,-1,-1,-1,-1; -1,0,-1,-1,0; -1,-1,0,0,0; -1,-1,0,1,0; -1,0,0,0,1]); vecsum((m^n)[1,]); \\ Michel Marcus, Jun 21 2017

a(n) = 2*a(n-1)+5*a(n-2)-13*a(n-3)+7*a(n-4)-a(n-5); a(0)=1, a(1)=-4, a(2)=6, a(3)=-24, a(4)=41 (follows from the minimal polynomial x^5-2*x^4-5*x^3+13*x^2-7*x+1 of the matrix M).

G.f.: (1-3*x^3+9*x^2-6*x)/(1+x^5-7*x^4+13*x^3-5*x^2-2*x). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

Edited by N. J. A. Sloane, Oct 29 2006

A227240 Numbers k such that sigma(k) divides sigma(2k) and sigma(2k + 1).

Original entry on oeis.org

1, 3, 5, 7, 11, 23, 29, 41, 53, 77, 83, 89, 103, 113, 131, 143, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 533, 593, 641, 653, 659, 667, 683, 719, 743, 761, 807, 809, 817, 911, 953, 1013, 1019, 1031, 1049, 1073, 1103, 1223, 1229, 1289, 1409

Offset: 1

Alex Ratushnyak, Jul 03 2013

nonn

Numbers such that 2*k and/or 2*k + 1 is also in the sequence: 1, 3, 5, 11, 41, 89, 179, 359, 509, 719, 743, ... (Cf. A007700).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000203, A007700, A058072, A058073.

Select[Range[1000], IntegerQ[DivisorSigma[1, 2#]/DivisorSigma[1, #]] && IntegerQ[DivisorSigma[1, 2# + 1]/DivisorSigma[1, #]] &] (* Alonso del Arte, Jul 15 2013 *)
Select[Range[1500],And@@Divisible[{DivisorSigma[1,2#],DivisorSigma[1,2#+1]}, DivisorSigma[1,#]]&] (* Harvey P. Dale, Feb 25 2016 *)

isok(n) = my(sn=sigma(n)); !(sigma(2*n) % sn) && !(sigma(2*n+1) % sn); \\ Michel Marcus, Oct 02 2017

A278932 Numbers n such that n remains prime through 6 iterations of function f(x) = 2x + 1.

Original entry on oeis.org

1122659, 2164229, 2329469, 10257809, 10309889, 12314699, 14030309, 14145539, 19099919, 23103659, 24176129, 28843649, 37088729, 38199839, 42389519, 49160099, 50785439, 52554569, 62800169, 68718059, 85864769, 88174049, 95831189, 105109139, 105388169

Offset: 1

John Cerkan, Dec 01 2016

nonn

n, 2*n+1, 4*n+3, 8*n+7, 16*n+15, 32*n+31, and 64*n+63 are primes.

a(n) == 29 (mod 30).

Subsequence of A007700, A023272, A023302, and A023330.

a005408(n) = 2*n+1
count(n) = my(k=n, i=0); while(ispseudoprime(k), k=a005408(k); i++); i
is(n) = count(n) > 6 \\ Felix Fröhlich, Dec 05 2016

cp's OEIS Frontend

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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A059688 Length of Cunningham chain containing prime(n) either as initial, internal or final term.

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A059767 Initial (unsafe) primes of Cunningham chains of first type with length exactly 7.

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A162019 Double-safe primes which are also double-Sophie Germain primes.

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A059690 Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.

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A122173 Expansion of -x * (x^5+x^4-15*x^3+19*x^2-8*x+1) / (x^6-12*x^5+34*x^4-30*x^3+6*x^2+3*x-1).

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A122174 First row sum of the matrix M^n, where M is the 5 X 5 matrix {{0,-1,-1,-1,-1}, {-1,0,-1,-1,0}, {-1,-1,0,0,0}, {-1,-1,0,1,0}, {-1,0,0,0,1}}.

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A122173 Expansion of -x * (x^5+x^4-15x^3+19x^2-8x+1) / (x^6-12x^5+34x^4-30x^3+6x^2+3x-1).

A227240 Numbers k such that sigma(k) divides sigma(2k) and sigma(2k + 1).