A046136 -id:A046136 - cp's OEIS frontend

A237890 Primes p such that p^2 + 4 and p^2 + 10 are also primes.

3, 7, 13, 97, 487, 613, 743, 827, 883, 1117, 1987, 2477, 2887, 3863, 4483, 5153, 5557, 5683, 5923, 5953, 6287, 7643, 7937, 8093, 9323, 10343, 12377, 13033, 13063, 14087, 14767, 15373, 16937, 17713, 17987, 18257, 19013, 19333, 19753, 19853, 20287, 20873, 21673

Offset: 1

K. D. Bajpai, Feb 15 2014

nonn

			7 is prime and appears in the sequence because 7^2+4 = 53 and 7^2+10 = 59 are also primes.
97 is prime and appears in the sequence because 97^2+4 = 9413 and 97^2+10 = 9419 are also primes.

K. D. Bajpai, Table of n, a(n) for n = 1..1300

Cf. A000040, A023200, A046136, A230223.

KD := proc() local a,b,d;  a:=ithprime(n);  b:=a^2+4; d:=a^2+10;  if isprime (b) and isprime(d) then RETURN (a); fi;  end: seq(KD(), n=1..5000);

Select[Prime[Range[5000]], PrimeQ[#^2 + 4] && PrimeQ[#^2 + 10] &]

s=[]; forprime(p=2, 25000, if(isprime(p^2+4) && isprime(p^2+10), s=concat(s, p))); s \\ Colin Barker, Feb 15 2014

A236302 Primes p such that p+8, p+86, p+864 are prime.

Original entry on oeis.org

23, 743, 983, 1163, 1373, 1613, 2663, 4013, 4643, 6113, 6863, 7583, 7673, 8513, 10313, 10853, 11243, 12503, 12713, 15233, 15263, 25733, 25763, 28703, 39623, 40763, 42743, 46133, 54623, 56093, 61643, 63353, 65003, 67733, 68813, 70373, 70913, 71933, 78893, 86453

Offset: 1

K. D. Bajpai, Apr 21 2014

nonn

All the terms in the sequence are congruent to 2 mod 3.

The constants in the definition (8, 86 and 864) are the concatenation of successive even digits 8,6 and 4.

			a(1) = 23 is a prime: 23+8 = 31, 23+86 = 109 and 23+864 = 887 are also prime.
a(2) = 743 is a prime: 743+8 = 751, 743+86 = 829 and 743+864 = 1607 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..4796

Cf. A000040, A023200, A046136, A230223, A237890.

KD:= proc() local a,b,d,e,f; a:= ithprime(n); b:=a+8;d:=a+86;e:=a+864; if isprime(b)and isprime(d)and isprime(e) then return (a) :fi; end: seq(KD(), n=1..15000);

KD = {}; Do[p = Prime[n];If[PrimeQ[p + 8] && PrimeQ[p + 86] && PrimeQ[p + 864],AppendTo[KD, p]], {n, 15000}]; KD
c=0; p=Prime[n]; Do[If[PrimeQ[p+8]&&PrimeQ[p+86]&&PrimeQ[p+864],c=c+1;Print[c,"  ",p]], {n,1,5*10^6}]; (*b-file*)

s=[]; forprime(p=2, 90000, if(isprime(p+8) && isprime(p+86) && isprime(p+864), s=concat(s, p))); s \\ Colin Barker, Apr 21 2014

A236304 Primes p such that p+12, p+1234 and p+123456 are also prime.

Original entry on oeis.org

127, 907, 3037, 3457, 5737, 7057, 11047, 15427, 15667, 21517, 24697, 30307, 38287, 38317, 39607, 40177, 46477, 47797, 48787, 51157, 52177, 57667, 65587, 70627, 70867, 71887, 72997, 74857, 75277, 80317, 99817, 100447, 103657, 106747, 128437, 130087, 132157

Offset: 1

K. D. Bajpai, Apr 21 2014

nonn

All the terms in the sequence are congruent to 1 mod 3.

The constants in the definition (12, 1234 and 123456) are the concatenation of digits 1,2,3,4,5 and 6.

			a(1) = 127 is a prime: 127+12 = 139, 127+1234 = 1361 and 127+123456 = 123583 are also prime.
a(2) = 907 is a prime: 907+12 = 919, 907+1234 = 2141 and 907+123456 = 124363 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5077

Cf. A000040, A023200, A046136, A230223, A236302, A237890.

KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+12;d:=a+1234;e:=a+123456; if isprime(b)and isprime(d)and isprime(e)  then return (a) :fi; end: seq(KD(), n=1..15000);

KD={}; Do[p=Prime[n]; If[PrimeQ[p+12]&&PrimeQ[p+1234]&&PrimeQ[p+123456], AppendTo[KD,p]], {n,15000}];KD
c=0; p=Prime[n]; Do[If[PrimeQ[p+12]&&PrimeQ[p+1234]&&PrimeQ[p+123456], c=c+1; Print[c,"  ",p]],{n,1,5*10^6}];(*b-file*)

s=[]; forprime(p=2, 140000, if(isprime(p+12) && isprime(p+1234) && isprime(p+123456), s=concat(s, p))); s \\ Colin Barker, Apr 22 2014

A241485 Primes p such that p+2, p+222 and p+2222 are also prime.

Original entry on oeis.org

17, 29, 59, 71, 281, 461, 827, 1151, 1277, 1289, 1487, 1667, 1877, 1931, 2687, 2789, 2801, 3359, 3557, 3851, 4049, 4229, 4259, 4481, 4649, 5417, 5519, 5657, 5867, 5879, 6089, 6131, 6299, 6359, 6779, 6791, 7127, 7211, 8291, 8837, 9719, 10067, 10937, 13397, 13679

Offset: 1

K. D. Bajpai, Apr 23 2014

nonn

All the terms in the sequence are congruent to 2 mod 3.

The constants in the definition (2, 222 and 2222) are the concatenation of digit 2.

			a(1) = 17 is a prime: 17+2 = 19, 17+222 = 239 and 17+2222 = 2239 are also prime.
a(2) = 29 is a prime: 29+2 = 31, 29+222 = 251 and 29+2222 = 2251 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A023200, A046136, A230223, A236302, A237890.

KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+2;d:=a+222;e:=a+2222; if isprime(b)and isprime(d)and isprime(e)  then return (a) :fi; end: seq(KD(), n=1..5000);

KD={}; Do[p=Prime[n];If[PrimeQ[p+2]&&PrimeQ[p+222]&&PrimeQ[p+2222], AppendTo[KD,p]], {n,5000}]; KD
(*For b-file*) c=0;p=Prime[n];Do[If[PrimeQ[p+2]&&PrimeQ[p+222]&&PrimeQ[p+2222],c=c+1; Print[c,"  ",p]],{n,1,3*10^6}];

A144594 Primes p such that p, p+4, p+10, p+22, p+24, p+42 are all primes.

Original entry on oeis.org

19, 37, 499, 1009, 1279, 1429, 2689, 5077, 13687, 16879, 17467, 23017, 25579, 32299, 33577, 41179, 48757, 85597, 92377, 120997, 125617, 128389, 143239, 152419, 159769, 324427, 327469, 351037, 352399, 422857, 473719, 499669, 518737, 519349

Offset: 1

Philip Mole (molep(AT)comcen.com.au), Jan 13 2009

nonn

Subsequence of A046136. - R. J. Mathar, Jan 14 2009

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

isA046136 := proc(n) if isprime(n) and isprime(n+4) and isprime(n+10) then true; else false; fi; end: isA144594 := proc(n) if isA046136(n) and isprime(n+22) and isprime(n+24) and isprime(n+42) then true; else false; fi; end: for n from 2 to 1000000 do if isA144594(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Jan 14 2009

lst={}; Do[p=Prime[n]; If[PrimeQ[p+4]&&PrimeQ[p+10]&&PrimeQ[p+22]&&PrimeQ[p+24]&&PrimeQ[p+42],AppendTo[lst,p]],{n,3*8!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Select[Prime[Range[44000]],AllTrue[#+{4,10,22,24,42},PrimeQ]&] (* Harvey P. Dale, Oct 02 2021 *)

Definition corrected by N. J. A. Sloane, Jan 13 2009
Inserted missing values R. J. Mathar, Jan 14 2009
More terms from Vladimir Joseph Stephan Orlovsky, Jan 15 2009

A235871 Primes p such that p+2, p+24 and p+246 are also primes.

Original entry on oeis.org

5, 17, 107, 617, 857, 1277, 1487, 2087, 3167, 3557, 4217, 6947, 7457, 7877, 10067, 12917, 13217, 14387, 15137, 17657, 20897, 21317, 22367, 22697, 27407, 27527, 27917, 28547, 29207, 29387, 30467, 31727, 32117, 33287, 33617, 35507, 36107, 47657, 49367, 49787

Offset: 1

K. D. Bajpai, Apr 21 2014

nonn

All the terms in the sequence are congruent to 5 mod 6.

The constants in the definition (2, 24 and 246) are the concatenation of first even digits 2,4 and 6.

			a(2) = 17 is a prime: 17+2 = 19, 17+24 = 41 and 17+246 = 263 are also prime.
a(3) = 107 is a prime: 107+2 = 119, 107+24 = 131 and 107+246 = 353 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5178

Cf. A000040, A023200, A046136, A230223, A237890.

KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+2;d:=a+24;e:=a+246; if isprime(b) and isprime(d) and isprime(e) then return (a) :fi; end: seq(KD(), n=1..15000);

KD = {}; Do[p = Prime[n]; If[PrimeQ[p + 2] && PrimeQ[p + 24] && PrimeQ[p + 246], AppendTo[KD, p]], {n, 15000}]; KD
c = 0; p = Prime[n]; Do[If[PrimeQ[p + 2] && PrimeQ[p + 24] && PrimeQ[p + 246], c = c + 1; Print[c, " ", Prime[n]]], {n, 1, 5000000}];  (* b - file *)

s=[]; forprime(p=2, 50000, if(isprime(p+2) && isprime(p+24) && isprime(p+246), s=concat(s, p))); s \\ Colin Barker, Apr 21 2014

A241486 Primes p such that p+4, p+444 and p+4444 are also prime.

Original entry on oeis.org

13, 19, 79, 103, 229, 307, 643, 853, 859, 937, 1087, 1213, 1297, 1423, 1567, 1609, 1867, 2347, 2389, 2473, 3163, 3463, 3919, 4003, 4153, 4783, 4969, 5077, 5347, 5413, 5479, 5647, 5689, 5857, 6733, 6907, 6967, 7933, 8269, 9277, 9337, 9463, 10687, 10729, 11083

Offset: 1

K. D. Bajpai, Apr 23 2014

nonn

All the terms in the sequence are congruent to 1 mod 6.

The constants in the definition (4, 444 and 4444) are the concatenations of the digit 4.

			a(1) = 13 is a prime: 13+4 = 17, 13+444 = 457 and 13+4444 = 4457 are also prime.
a(2) = 19 is a prime: 19+4 = 23, 19+444 = 463 and 19+4444 = 4463 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A023200, A046136, A230223, A236302, A236304, A237890, A241485.

KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+4; d:=a+444; e:=a+4444;if isprime(b)and isprime(d)and isprime(e)then return (a): fi;  end: seq(KD(), n=1..5000);

KD = {}; Do[p = Prime[n]; If[PrimeQ[p + 4] && PrimeQ[p + 444] && PrimeQ[p + 4444], AppendTo[KD, p]], {n, 5000}]; KD
(* For the b-file*) c = 0; p = Prime[n]; Do[If[PrimeQ[p + 4] && PrimeQ[p + 444] && PrimeQ[p + 4444], c = c + 1; Print[c, "  ", p]], {n, 1, 3*10^6}];

s=[]; forprime(p=2, 12000, if(isprime(p+4) && isprime(p+444) && isprime(p+4444), s=concat(s, p))); s \\ Colin Barker, Apr 25 2014

A241487 Primes p such that p+6, p+666 and p+6666 are also prime.

Original entry on oeis.org

7, 53, 67, 157, 191, 311, 331, 347, 353, 373, 443, 563, 571, 641, 821, 823, 857, 941, 1033, 1087, 1123, 1283, 1423, 1607, 1621, 1873, 1997, 2011, 2137, 2333, 2383, 2543, 2657, 2677, 2797, 2957, 3301, 3511, 3607, 3671, 3691, 3797, 3847, 4133, 5113, 5147, 5231

Offset: 1

K. D. Bajpai, Apr 23 2014

nonn

The constants in the definition (6, 666 and 6666) are concatenations of the digit 6.

			a(2) = 53 is a prime: 53+6 = 59, 53+666 = 719 and 53+6666 = 6719 are also prime.
a(3) = 67 is a prime: 67+6 = 73, 67+666 = 733 and 67+6666 = 6733 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A023200, A046136, A230223, A236302, A236304, A237890, A241485.

KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+2;d:=a+222;e:=a+2222; if isprime(b)and isprime(d)and isprime(e)  then return (a) :fi; end: seq(KD(), n=1..5000);

KD = {}; Do[p = Prime[n];If[PrimeQ[p + 6] && PrimeQ[p + 666] && PrimeQ[p + 6666],AppendTo[KD, p]], {n, 5000}]; KD
(*For the b-file*) c = 0; p = Prime[n]; Do[If[PrimeQ[p + 6] && PrimeQ[p + 666] && PrimeQ[p + 6666], c = c + 1;Print[c, "  ", p]], {n, 1, 2*10^6}];

s=[]; forprime(p=2, 6000, if(isprime(p+6) && isprime(p+666) && isprime(p+6666), s=concat(s, p))); s \\ Colin Barker, Apr 25 2014

A241488 Primes p such that p+8, p+888 and p+8888 are also prime.

Original entry on oeis.org

53, 263, 389, 431, 983, 1013, 1061, 1223, 1571, 1823, 2789, 3323, 3533, 3911, 4211, 5849, 6563, 6653, 7019, 7481, 8369, 8963, 9041, 9173, 9413, 9539, 9803, 10091, 10559, 10979, 12611, 12689, 12911, 13163, 13751, 13781, 14243, 14879, 15083, 16691, 17231, 17483

Offset: 1

K. D. Bajpai, Apr 23 2014

nonn

All the terms in the sequence are congruent to 2 mod 3.

The constants in the definition (8, 888 and 8888) are the concatenation of digit 8.

			a(1) = 53 is a prime: 53+8 = 61, 53+888 = 941 and 53+8888 = 8941 are also prime.
a(2) = 263 is a prime: 263+8 = 271, 263+888 = 1151 and 263+8888 = 9151 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A023200, A046136, A230223, A236302, A236304, A237890, A241485.

KD:= proc() local a,b,d,e; a:= ithprime(n); b:=a+8;d:=a+888;e:=a+8888; if isprime(b)and isprime(d)and isprime(e)  then return (a) :fi; end: seq(KD(), n=1..5000);

KD = {}; Do[p = Prime[n]; If[PrimeQ[p + 8] && PrimeQ[p + 888] && PrimeQ[p + 8888], AppendTo[KD, p]], {n, 5000}]; KD
(*For the b-file*)  c = 0; p = Prime[n]; Do[If[PrimeQ[p + 8] && PrimeQ[p + 888] && PrimeQ[p + 8888], c = c + 1; Print[c, "  ", p]], {n, 1, 5*10^6}];
Select[Prime[Range[2500]],AllTrue[#+{8,888,8888},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 31 2017 *)

s=[]; forprime(p=2, 18000, if(isprime(p+8) && isprime(p+888) && isprime(p+8888), s=concat(s, p))); s \\ Colin Barker, Apr 25 2014

A096659 Earliest value of C in y = x^2+(2n-1)x+C such that y is prime for all x = 0 to n.

Original entry on oeis.org

2, 3, 3, 5, 23, 31, 47, 59, 13, 29, 17, 37, 23, 47, 73, 251, 281, 313, 347, 383, 421

Offset: 0

Ray G. Opao, Aug 25 2004

a(0) is the first term of A000040; a(1) is the first term of A001359; a(2) is the first term of A046136; a(3) is the first term of A097434; a(4) is the first term of A097436; a(5) is the first term of A097437; a(6) is the first term of A097458; a(7) is the first term of A097459; a(8) is the first term of A097460; a(9) is the first term of A097461.
Sequence is infinite under Dickson's conjecture. - Charles R Greathouse IV, Aug 11 2013
a(20) > 10^10; probably a(20) > 10^20. - Charles R Greathouse IV, Aug 12 2013

			Triangle of y, primes, starts:
2,
3, 7,
3, 9, 17,
5, 13, 23, 35,
23, 33, 45, 59, 75,
31, 43, 57, 73, 91, 111,
47, 61, 77, 95, 115, 137, 161,
59, 75, 93, 113, 135, 159, 185, 213,
...

a(n) = C = 1; ok = 0; while (! ok, ok = 1; for (x = 0, n, if (! isprime(x^2+(2*n-1)*x+C), ok = 0; break;);); if (ok, return (C)); C++;); \\ Michel Marcus, Aug 10 2013

works(C,n)=for(x=1,n,if(!isprime(x^2+(2*n-1)*x+C), return(0)));1
a(n)=forprime(C=2,,if(works(C,n),return(C))) \\ Charles R Greathouse IV, Aug 12 2013

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A237890 Primes p such that p^2 + 4 and p^2 + 10 are also primes.

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A236302 Primes p such that p+8, p+86, p+864 are prime.

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A236304 Primes p such that p+12, p+1234 and p+123456 are also prime.

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A241485 Primes p such that p+2, p+222 and p+2222 are also prime.

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A144594 Primes p such that p, p+4, p+10, p+22, p+24, p+42 are all primes.

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A235871 Primes p such that p+2, p+24 and p+246 are also primes.

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A241486 Primes p such that p+4, p+444 and p+4444 are also prime.

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