A023287
Primes that remain prime through 3 iterations of function f(x) = 6x + 1.
Original entry on oeis.org
61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341
Offset: 1
First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.
-
[n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // Vincenzo Librandi, Aug 04 2010
-
k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by Zak Seidov, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&,n ,3]],PrimeQ]; Select[Prime[Range[9000]],thrQ] (* Harvey P. Dale, Mar 03 2024 *)
A023347
Primes which remain prime through 5 iterations of function f(x) = 8x + 1.
Original entry on oeis.org
831167, 1154567, 2502767, 3019787, 3675197, 5056577, 6352487, 14519177, 26724377, 43003577, 47378927, 47695607, 56406197, 86332457, 86611757, 99568757, 121967987, 126435527, 127990997, 128149127, 128975057, 145281557, 155715407
Offset: 1
First chain is {831167, 6649337, 53194697, 425557577, 3404460617, 27235684937};
If p is congruent to {1,3,7,9} mod 10, then consecutive iterates are congruent to {9,5,7,3}, {3,1,7,5}, {5,9,7,1} respectively; so only 10k+7 may remain prime through five iterations, as sequence demonstrates nicely. - _Labos Elemer_, Jul 23 2003
-
k=0; m=8; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
it5Q[n_]:=AllTrue[Rest[NestList[8#+1&,n,5]],PrimeQ]; Select[Prime[Range[ 9*10^6]],it5Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2014 *)
A086127
Numbers k such that k remains prime after five iteration of function f(j) = 14*f(j)+1, starting at f(1) = prime.
Original entry on oeis.org
4889, 18059, 62639, 225527, 557093, 604973, 700703, 804077, 806903, 837077, 1341203, 1363403, 1932197, 2004269, 2062703, 2284637, 2797463, 3157379, 3493103, 3746399, 3995687, 4155413, 4227893, 4493297, 5534939, 5708603
Offset: 1
First chain is: {4889,68447,958259,13415627,187818779,2629462907}.
10th chain is {837077,11719079,164067107,2296939499,32157152987,450200141819}.
-
k=0; m=14; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
Select[Range[6000000],And@@PrimeQ[NestList[14#+1&,#,5]]&] (* Harvey P. Dale, Sep 17 2012 *)
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
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.
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.
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
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.
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
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).
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.
Showing 1-6 of 6 results.
Comments