A059455 -id:A059455 - cp's OEIS frontend

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

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

A152294 Primes of the form : (p-n)/(n+1)=prime and (n+1)*p+n=prime. n=4.

Original entry on oeis.org

29, 89, 419, 509, 659, 1259, 1289, 1319, 1949, 2099, 2309, 2339, 2609, 2939, 3989, 4049, 6089, 6599, 7559, 8609, 9239, 9539, 10709, 12659, 12899, 13469, 13499, 18119, 20399, 21089, 21269, 21419, 22469, 23369, 26669, 27539, 28559, 30059, 30449

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 02 2008

nonn

This is the general form : (p-n)/(n+1)=primeand(n+1)*p+n=prime; 'Safe' primes and'Sophie Germain' primes just one part of this general form; If n=1 then we got'Safe' primes and'Sophie Germain' primes.

Cf. A059455, A152292, A152293.

lst={};n=4;Do[p=Prime[k];If[PrimeQ[(p-n)/(n+1)]&&PrimeQ[(n+1)*p+n],AppendTo[lst,p]],{k,7!}];lst

A152295 Primes of the form : (p-n)/(n+1)=prime and (n+1)*p+n=prime. n=5.

Original entry on oeis.org

17, 71, 83, 107, 191, 227, 251, 263, 431, 443, 479, 503, 587, 827, 839, 911, 983, 1091, 1151, 1163, 1187, 1619, 1667, 1847, 1907, 2087, 2243, 2459, 2591, 3023, 3467, 4463, 4871, 4943, 5471, 5519, 5651, 5807, 5903, 6131, 6203, 6299, 6311, 6563, 6983, 7127

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 02 2008

nonn

This is the general form : (p-n)/(n+1)=primeand(n+1)*p+n=prime; 'Safe' primes and'Sophie Germain' primes just one part of this general form; If n=1 then we got'Safe' primes and'Sophie Germain' primes.

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A059455, A152292, A152293, A152294

lst={};n=5;Do[p=Prime[k];If[PrimeQ[(p-n)/(n+1)]&&PrimeQ[(n+1)*p+n],AppendTo[lst,p]],{k,7!}];lst
Select[Prime[Range[1000]],AllTrue[{(#-5)/6,6#+5},PrimeQ]&] (* This program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 29 2014 *)

A156658 Primes p such that also 2*p+1 or (p-1)/2 is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 47, 53, 59, 83, 89, 107, 113, 131, 167, 173, 179, 191, 227, 233, 239, 251, 263, 281, 293, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 509, 563, 587, 593, 641, 653, 659, 683, 719, 743, 761, 809, 839, 863, 887, 911, 953, 983, 1013

Offset: 1

Reinhard Zumkeller, Feb 13 2009

nonn

Union of A005384 and A005385;
The intersection of A005384 and A005385 is given by A059455.
A156660(a(n)) + A156659(a(n)) > 0;
primes occurring in Cunningham chains of the first kind.
A156876 gives the number of these numbers <= n. [Reinhard Zumkeller, Feb 18 2009]

Robert Price, Table of n, a(n) for n = 1..14198
Wikipedia, Sophie Germain prime
Wikipedia, Safe prime
Wikipedia, Cunningham chain

Cf. A005384, A005385, A059500, A156659, A156660, A156876.

select(t -> isprime(t) and (isprime(2*t+1) or isprime((t-1)/2)), [2,seq(p,p=3..10000,2)]); # Robert Israel, May 03 2016

Select[Prime@ Range@ 180, PrimeQ[2 # + 1] || PrimeQ[(# - 1)/2] &] (* Michael De Vlieger, Apr 06 2016 *)

lista(nn) = {forprime(p=2, nn, if (isprime(2*p+1) || isprime((p-1)/2), print1(p, ", ")););} \\ Michel Marcus, Apr 06 2016

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

A100930 Semiprimes of the form (2p+1)(p-1)/2, p prime.

Original entry on oeis.org

22, 115, 517, 6847, 31951, 128701, 516601, 1037851, 2070001, 4156501, 4254937, 6045451, 7945351, 8425957, 8777887, 9137017, 13124317, 14278951, 14460907, 14920837, 24194101, 29146501, 31795501, 47592751, 48758797, 50108701

Offset: 1

Reinhard Zumkeller, Nov 22 2004

nonn

a(n) = (2*A059455(n)^2 - A059455(n) - 1) / 2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Sophie Germain Prime

Subsequence of A100929.

Cf. A001358, A005384, A005385.

Select[Table[((2p+1)(p-1))/2,{p,Prime[Range[2000]]}],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 13 2020 *)

A075705 Safe primes (A005385) (p and (p-1)/2 are primes) such that 6*p+1 is also prime.

Original entry on oeis.org

5, 7, 11, 23, 47, 83, 107, 263, 347, 467, 503, 863, 887, 1283, 1487, 1823, 1907, 2027, 2063, 2447, 2903, 3203, 3623, 4007, 4127, 4547, 4703, 4787, 5387, 5807, 7523, 7703, 8147, 8423, 11423, 11483, 11807, 12107, 12227, 12647, 12983, 13043, 13163, 14087, 14207

Offset: 1

Jani Melik, Oct 02 2002

nonn

			47 is prime, so is (47-1)/2=23 and also 6*47+1=283; 83 is a prime, (83-1)/2=41 and 6*83+1=499, ...

David Consiglio, Jr. and T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A059455, A007693.

ts_sg_var_pras := proc(nmax) local i,tren,atek; tren := 0: for i from 1 to nmax do atek := numtheory[safeprime](i): if (atek > tren) then if (isprime(atek)='true' and isprime(6*atek+1)='true') then tren := atek: fi; fi; od; end: seq(ts_sg_var_pras(i), i=1..3000);

Select[Range[20000], PrimeQ[#] && PrimeQ[(#-1)/2] && PrimeQ[6#+1] &] (* T. D. Noe, Nov 07 2011 *)
Select[Prime[Range[1700]],And@@PrimeQ[{(#-1)/2,6#+1}]&] (* Harvey P. Dale, Feb 28 2013 *)

A075706 Safe primes (A005385) (p and (p-1)/2 are primes) such that 8*p+1 (A023228) is also prime.

Original entry on oeis.org

5, 11, 107, 179, 347, 479, 1187, 1307, 1367, 1487, 1619, 2027, 2207, 2447, 2999, 3119, 3467, 4007, 4079, 4139, 4799, 5087, 5807, 5927, 5939, 6827, 7079, 7247, 8699, 9587, 9839, 10607, 12107, 12539, 12659, 14207, 15299, 16139, 16187, 17027

Offset: 1

Jani Melik, Oct 02 2002

nonn

			11 is a prime, so is (11-1)/2=5 and also 8*11+1=89; 107 is a prime, (107-1)/2=53 and 8*107+1=857, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A005384, A005385, A059455, A023228.

ts_sg8_var_pras := proc(nmax) local i,tren,atek; tren := 0: for i from 1 to nmax do atek := numtheory[safeprime](i): if (atek > tren) then if (isprime(atek)='true' and isprime(6*atek+1)='true') then tren := atek: fi; fi; od; end: seq(ts_sg8_var_pras(i), i=1..3000);

Select[Prime[Range[2000]],AllTrue[{(#-1)/2,8#+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 31 2020 *)

forprime(p=3,20000,if(isprime((p-1)/2),if(isprime(8*p+1),print1(p","))))

More terms from Ralf Stephan, Mar 19 2003

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

cp's OEIS Frontend

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

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A152294 Primes of the form : (p-n)/(n+1)=prime and (n+1)*p+n=prime. n=4.

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A152295 Primes of the form : (p-n)/(n+1)=prime and (n+1)*p+n=prime. n=5.

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A156658 Primes p such that also 2*p+1 or (p-1)/2 is prime.

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

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A100930 Semiprimes of the form (2*p+1)*(p-1)/2, p prime.

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A075705 Safe primes (A005385) (p and (p-1)/2 are primes) such that 6*p+1 is also prime.

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Maple

Mathematica

A075706 Safe primes (A005385) (p and (p-1)/2 are primes) such that 8*p+1 (A023228) is also prime.

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Maple

Mathematica

A100930 Semiprimes of the form (2p+1)(p-1)/2, p prime.

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).