A059455 -id:A059455 - cp's OEIS frontend

A158021 Primes such that p, 10*p-1 and (p-1)/2 are all prime.

11, 23, 83, 107, 167, 179, 227, 347, 479, 587, 839, 863, 983, 1283, 1367, 1439, 1487, 1619, 1823, 1907, 2027, 2039, 2447, 2879, 2963, 2999, 3119, 3203, 3623, 3863, 4127, 4139, 4259, 4283, 4787, 5099, 5483, 5879, 6719, 6779, 6983, 7247, 7703, 7727, 7823

Offset: 1

Roger L. Bagula and Gary W. Adamson, Mar 11 2009

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A059455. Intersection of A000040 and A158019.

filter:= proc(n) isprime(n) and isprime((n-1)/2) and isprime(10*n-1) end proc:
select(filter, [seq(i,i=3 .. 10000, 4)]); # Robert Israel, Dec 12 2024

Flatten[Table[If[PrimeQ[n] && PrimeQ[10*n - 1] && PrimeQ[( n - 1)/2], n, {}], {n, 1, 10000}]]

A075707 Safe primes (A005385) (p and (p-1)/2 are primes) such that 12*p+1 is also prime.

Original entry on oeis.org

5, 23, 59, 83, 383, 479, 503, 719, 839, 863, 1619, 2039, 2099, 2579, 2819, 2879, 3023, 4139, 4259, 4679, 4703, 4919, 5879, 6719, 6779, 7559, 8039, 8783, 8819, 10799, 11279, 11423, 12203, 12659, 12899, 12983, 13523, 13799, 14159, 14303, 14699, 15683, 18119, 18443, 19259, 19379, 20183, 20663, 21059, 23663, 24083, 24239, 24659, 27239, 28163, 29123, 29339, 29483, 29759, 30803, 31139, 31583, 36923, 37463, 38603, 39119, 39503, 39839, 39983, 41879, 42299, 42443, 43403, 44519, 44939, 46679, 47339, 47819, 47963

Offset: 1

Jani Melik, Oct 02 2002

nonn

			23 is a prime, so is (23-1)/2=11 and also 12*23+1=277, 59 is a prime, (59-1)/2=29 and 12*59+1=709, ...

Cf. A005384, A005385, A059455.

ts_sgB_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_sgB_var_pras(i), i=1..3000);

okQ[n_]:=PrimeQ[(n-1)/2]&&PrimeQ[12n+1]
Select[Prime[Range[5000]],okQ] (* Harvey P. Dale, Nov 21 2010 *)

More terms from Harvey P. Dale, Nov 21 2010

A085832 Smallest prime p such that both (p-1)/2^(2n-1) and 2^(2n-1)*p+1 are primes.

Original entry on oeis.org

5, 17, 1889, 21377, 183809, 83969, 40961, 79003649, 245235713, 5767169, 1004535809, 897581057, 41238396929, 13555990529, 2357400174593, 3438121320449, 12360915877889, 188188287041537, 286010462175233

Offset: 1

Robert G. Wilson v, Jul 26 2003

nonn

I have found no primes less than the 10^7th prime where an even power of 2 results in the two prime forms above simultaneously except for 2^2 and p=13.

Cf. A059455, A085958, A086476, A086477, A086478.

f[n_] := Block[{k = 1}, While[ !PrimeQ[k] || !PrimeQ[(k - 1)/2^n] || !PrimeQ[2^n*k + 1], k += 2^n]; k]; Table[ f[n], {n, 1, 37, 2}]

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

A110092 Smallest prime ending (through <2+1> or <2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

Original entry on oeis.org

17, 59, 73, 4079, 47, 2879, 1065601

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 not be involved in a chain of the other kind (i.e. the chain may not be connected to another one).

			a(1)=17 because 2, 3, 5, 7, 11 and 13 are part of longer chains whatever the operator; 17 is the first completely isolated prime.
a(2)=59 because it ends the first two primes chain not connected to another one: 29->59.

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, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Terms computed by Gilles Sadowski.

A110093 Smallest prime ending (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, 7, 5, 4079, 47, 2879, 1065601

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); but 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)=11 because 2, 3, 5 and 7 are not ending chains; or are part of chains longer than one prime; 11, although is part of a five primes <2p+1> chain, is isolated through <2p-1>.
a(2)=7 because 7 ends through <2p+1> the first two primes chain: 3->7 (even if both primes are also part of <2p-1> chains).

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, A110059, A110056, A110038, A059766, A110027, A059764, A110025, A110024, A059763, A110022, A109998, A109946, A109927, A109835, A005603.

Terms computed by Gilles Sadowski.

cp's OEIS Frontend

A158021 Primes such that p, 10*p-1 and (p-1)/2 are all prime.

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A075707 Safe primes (A005385) (p and (p-1)/2 are primes) such that 12*p+1 is also prime.

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A085832 Smallest prime p such that both (p-1)/2^(2n-1) and 2^(2n-1)*p+1 are primes.

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Mathematica

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.

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A110092 Smallest prime ending (through <*2+1> or <*2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A110093 Smallest prime ending (through <*2+1> or/and <*2-1>) a complete Cunningham chain (of the first or the second kind) of length n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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.

A110092 Smallest prime ending (through <2+1> or <2-1> separately) a complete Cunningham chain (of the first or the second kind) of length n.

A110093 Smallest prime ending (through <2+1> or/and <2-1>) a complete Cunningham chain (of the first or the second kind) of length n.