A187058 -id:A187058 - cp's OEIS frontend

A187060 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..7.

11, 17, 41, 21557, 26681, 128981, 844427, 2073347, 3992201, 4889237, 6184637, 11900501, 21456047, 24598361, 33771581, 34864211, 50943791, 51448361, 51867197, 55793951, 56421347, 61218251, 67787537, 69726647, 76345121

Offset: 1

Jonathan Vos Post, Mar 03 2011

nonn

From Weber, p. 15. However, erroneous.

All terms = {11,17} mod 30. - Zak Seidov, May 08 2011

			a(4) <> 21577 because 0^2 + 0 + 21577 = 21577; 1^2 + 1 + 21577 = 21579 = 3 * 7193 thus exposing an error in Weber's paper; 2^2 + 2 + 21577 = 21583 = 113 * 191; 3^2 + 3 + 21577 = 21589 is prime; 4^2 + 4 + 21577 = 21597 = 3 * 23 * 313; 5^2 + 5 + 21577 = 21607 = 17 * 31 * 41 (a "3-brilliant number" rather than a prime); 6^2 + 6 + 21577 = 21619 = 13 * 1663; 7^2 + 7 + 21577 = 21633 = 3 * 7211.

Zak Seidov and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (Zak Seidov found the first 400 terms)
H. J. Weber, Regularities of Twin, Triplet and Multiplet Prime Numbers, arXiv:1103.0447 [math.NT], 2011-2012.

Cf. A187057, A187058, A144051.

okQ[n_] := And @@ PrimeQ[Table[i^2 + i + n, {i, 0, 7}]]; Select[Range[10000], okQ] (* T. D. Noe, Mar 03 2011 *)

for(k=1,50000,p=prime(k); if(isprime(p+2) && isprime(p+6) && isprime(p+12) && isprime(p+20) && isprime(p+30) && isprime(p+42) && isprime(p+56),print(p),)) \\ Nathaniel Johnston, Apr 26 2011

p=2;q=3;forprime(r=5,1e6,if(r-p==6 && q-p==2 && isprime(p+12) && isprime(p+20) && isprime(p+30) && isprime(p+42) && isprime(p+56),print(p));p=q;q=r) \\ Charles R Greathouse IV, Mar 04 2012

a(12)-a(25) from Nathaniel Johnston, Apr 26 2011

A144051 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..6.

Original entry on oeis.org

11, 17, 41, 1277, 21557, 26681, 28277, 113147, 128981, 421697, 665111, 844427, 1164587, 1615631, 2073347, 2798921, 2846771, 3053747, 3992201, 4889237, 5071667, 5093507, 5344247, 5706641, 6184637, 6383051, 8396777

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 08 2008

nonn

All terms = {11,17} mod 30. - Zak Seidov, May 08 2011

			a(3) = 41 because 0^2 + 0 + 41 = 41; 1^2 + 1 + 41 = 43; 2^2 + 2 + 41 = 47; 3^2 + 3 + 41 = 53; 4^2 + 4 + 41 = 61; 5^2 + 5 + 41 = 71; 6^2 + 6 + 41 = 83, all primes.

Zak Seidov, Table of n, a(n) for n = 1..1000
H. J. Weber, Regularities of Twin, Triplet and Multiplet Prime Numbers, arXiv:1103.0447 [math.NT], 2011-2012.

Cf. A187057, A187058.

lst={}; Do[p1=Prime[n]; If[PrimeQ[p2=p1+2] && PrimeQ[p3=p1+6] && PrimeQ[p4=p1+12] && PrimeQ[p5=p1+20] && PrimeQ[p6=p1+30] && PrimeQ[p7=p1+42], AppendTo[lst,p1]], {n,10^5}]; lst
okQ[n_] := And @@ PrimeQ[Table[i^2 + i + n, {i, 0, 6}]]; Select[Range[10000], okQ] (* T. D. Noe, Mar 03 2011 *)

A190817 Initial primes of 6 consecutive primes with consecutive gaps 2,4,6,8,10.

Original entry on oeis.org

13901, 21557, 28277, 55661, 68897, 128981, 221717, 354371, 548831, 665111, 954257, 1164587, 1246367, 1265081, 1538081, 1595051, 1634441, 2200811, 2798921, 2858621, 3053747, 3407081, 3414011, 3967487, 3992201, 4480241, 4608281, 4701731, 4809251, 5029457

Offset: 1

Zak Seidov, May 21 2011

nonn

a(1) = 13901 = A190814(5) = A187058(7) = A078847(24).

a(n) + 30 is the greatest term in the sequence of 6 consecutive primes with consecutive gaps 2, 4, 6, 8, 10. - Muniru A Asiru, Aug 10 2017

			For n = 1, 13901 is in the sequence because 13901, 13903, 13907, 13913, 13921, 13931 are consecutive primes and for n = 2, 21557 is in the sequence since 21557, 21559, 21563, 21569, 21577, 21587 are consecutive primes. - _Muniru A Asiru_, Aug 24 2017

Zak Seidov, Table of n, a(n) for n = 1..6000
R. J. Mathar, Table of Prime Gap Constellations

Cf. A078847, A187058, A190814, A190819, A190838.

K:=3*10^7+1;; # to get all terms <= K.
P:=Filtered([1,3..K],IsPrime);; I:=[2,4,6,8,10];;
P1:=List([1..Length(P)-1],i->P[i+1]-P[i]);;
P2:=List([1..Length(P)-Length(I)],i->[P1[i],P1[i+1],P1[i+2],P1[i+3],P1[i+4]]);;
P3:=List(Positions(P2,I),i->P[i]);  # Muniru A Asiru, Aug 24 2017

N:=10^7: # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+30,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],Primes[t+3]-Primes[t+2], Primes[t+4]-Primes[t+3], Primes[t+5]-Primes[t+4]]=[2,4,6,8,10], [$1..nops(Primes)-5])]; # Muniru A Asiru, Aug 04 2017

d = Differences[Prime[Range[100000]]]; Prime[Flatten[Position[Partition[d, 5, 1], {2, 4, 6, 8, 10}]]] (* T. D. Noe, May 23 2011 *)
With[{s = Differences@ Prime@ Range[10^6]}, Prime[SequencePosition[s, Range[2, 10, 2]][[All, 1]] ] ] (* Michael De Vlieger, Aug 16 2017 *)

lista(nn) = forprime(p=13901, nn, if(nextprime(p+1)==p+2 && nextprime(p+3)==p+6 && nextprime(p+7)==p+12 && nextprime(p+13)==p+20 && nextprime(p+21)==p+30, print1(p", "))); \\ Altug Alkan, Aug 16 2017

Additional cross references from Harvey P. Dale, May 10 2014

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

Original entry on oeis.org

11, 17, 41, 844427, 51448361, 86966771, 122983031, 180078317, 960959381, 1278189947, 1761702947, 1829187287, 2426256797, 2911675511, 3013107257, 4778888351, 5221343711

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..2713

Generates primes for x=1..k: A001359 (1), A022004 (2), A172454 (3), A187057 (4), A187058 (5), A144051 (6), A187060 (7), A190800 (8), this sequence (9), A191457 (10), A191458 (11), A253592 (12), A253605 (13). Each is by definition a subsequence of preceding sequences.
Subsequence such that x=10 gives a composite number: A211238.
Cf. A160548, A164926.

is(n)=for(x=0,9, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A247949 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...5.

Original entry on oeis.org

7, 43, 79, 457, 877, 967, 1093, 2437, 2683, 3187, 5077, 5923, 7933, 8233, 11923, 12889, 15787, 17389, 19993, 31543, 41113, 41617, 42457, 71359, 77863, 80683, 91393, 101719, 102643, 105967, 107347, 120163, 129733, 137593, 151783, 170263, 175723, 197569, 210127

Offset: 1

K. D. Bajpai, Jan 11 2015

nonn

All terms == 1 mod 6. - Robert Israel, Jan 11 2015

			a(1) = 7:
0^4 + 0^3 + 0^2 + 0 + 7 = 7;
1^4 + 1^3 + 1^2 + 1 + 7 = 11;
2^4 + 2^3 + 2^2 + 2 + 7 = 37;
3^4 + 3^3 + 3^2 + 3 + 7 = 127;
4^4 + 4^3 + 4^2 + 4 + 7 = 347;
5^4 + 5^3 + 5^2 + 5 + 7 = 787;
all six are primes.

K. D. Bajpai, Table of n, a(n) for n = 1..6730

Cf. A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458.

select(p -> andmap(isprime, [p, p+4, p+30, p+120, p+340, p+780]), [seq(6*i+1, i=1..10^5)]); # Robert Israel, Jan 11 2015

Select[f=k^4 + k^3 + k^2 + k; k = {0, 1, 2, 3, 4, 5}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[20000]],AllTrue[#+{4,30,120,340,780},PrimeQ]&] (* Harvey P. Dale, Dec 24 2023 *)

forprime(p=1, 500000, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780), print1(p,", ")))

A247966 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...6.

Original entry on oeis.org

43, 457, 967, 1093, 5923, 8233, 11923, 15787, 41113, 80683, 151783, 210127, 213943, 294919, 392737, 430879, 495559, 524827, 537007, 572629, 584557, 711727, 730633, 731593, 1097293, 1123879, 1138363, 1149163, 1396207, 1601503, 1739557, 1824139, 2198407, 2223853

Offset: 1

K. D. Bajpai, Jan 11 2015

nonn

			a(1) = 43:
0^4 + 0^3 + 0^2 + 0 + 43 = 43;
1^4 + 1^3 + 1^2 + 1 + 43 = 47;
2^4 + 2^3 + 2^2 + 2 + 43 = 73;
3^4 + 3^3 + 3^2 + 3 + 43 = 163;
4^4 + 4^3 + 4^2 + 4 + 43 = 383;
5^4 + 5^3 + 5^2 + 5 + 43 = 823;
6^4 + 6^3 + 6^2 + 6 + 43 = 1597;
all seven are primes.

K. D. Bajpai, Table of n, a(n) for n = 1..1405

Cf. A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458.

Select[f=k^4 + k^3 + k^2 + k; k = {0, 1, 2, 3, 4, 5, 6}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[200000]],AllTrue[#+{4,30,120,340,780,1554},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 10 2017 *)

forprime(p=1, 1e6, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780)&  isprime(p+1554), print1(p,", ")))

A248206 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...7.

Original entry on oeis.org

43, 457, 967, 11923, 15787, 41113, 213943, 294919, 392737, 430879, 524827, 572629, 730633, 1097293, 1149163, 2349313, 2738779, 3316147, 3666007, 5248153, 5396617, 5477089, 7960009, 9949627, 10048117, 11260237, 11613289, 15281023, 16153279, 17250367, 18733807

Offset: 1

K. D. Bajpai, Jan 11 2015

nonn

			a(1) = 43:
0^4 + 0^3 + 0^2 + 0 + 43 = 43;
1^4 + 1^3 + 1^2 + 1 + 43 = 47;
2^4 + 2^3 + 2^2 + 2 + 43 = 73;
3^4 + 3^3 + 3^2 + 3 + 43 = 163;
4^4 + 4^3 + 4^2 + 4 + 43 = 383;
5^4 + 5^3 + 5^2 + 5 + 43 = 823;
6^4 + 6^3 + 6^2 + 6 + 43 = 1597;
7^4 + 7^3 + 7^2 + 7 + 43 = 2843;
all eight are primes.

K. D. Bajpai, Table of n, a(n) for n = 1..511

Cf. A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458.

Select[f=k^4+k^3+k^2+k; k={0,1,2,3,4,5,6,7}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[12*10^5]],AllTrue[#+{4,30,120,340,780,1554,2800},PrimeQ]&] (* Harvey P. Dale, Apr 24 2022 *)

forprime(p=1, 1e8, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780)&  isprime(p+1554)& isprime(p+2800), print1(p,", ")))

A253915 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields primes for k = 0..8, but not for k = 9.

Original entry on oeis.org

43, 967, 11923, 213943, 2349313, 3316147, 30637567, 33421159, 39693817, 49978447, 105963769, 143405887, 148248949, 153756073, 156871549, 172981279, 187310803, 196726693, 203625283, 211977523, 220825453, 268375879, 350968543, 357834283, 414486697, 427990369

Offset: 1

K. D. Bajpai, Jan 18 2015

nonn

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

			a(1) = 43:
0^4 + 0^3 + 0^2 + 0 + 43 =   43;
1^4 + 1^3 + 1^2 + 1 + 43 =   47;
2^4 + 2^3 + 2^2 + 2 + 43 =   73;
3^4 + 3^3 + 3^2 + 3 + 43 =  163;
4^4 + 4^3 + 4^2 + 4 + 43 =  383;
5^4 + 5^3 + 5^2 + 5 + 43 =  823;
6^4 + 6^3 + 6^2 + 6 + 43 = 1597;
7^4 + 7^3 + 7^2 + 7 + 43 = 2843;
8^4 + 8^3 + 8^2 + 8 + 43 = 4723;
all nine are primes, and
9^4 + 9^3 + 9^2 + 9 + 43 = 7423 = 13*571 is composite.
The next prime for p=43 appears for k=13, namely 30983.

Jon E. Schoenfield, Table of n, a(n) for n = 1..155 (terms < 2*10^10)

Cf. A027445, A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458, A247949, A247966, A248206.

Select[Prime[Range[118*10^5]],AllTrue[#+{0,4,30,120,340,780,1554,2800,4680},PrimeQ]&&CompositeQ[#+7380]&] (* Harvey P. Dale, Sep 10 2021 *)

forprime(p=1, 1e10, if(isprime(p+4)&& isprime(p+30)&& isprime(p+120)&& isprime(p+340)&& isprime(p+780)&& isprime(p+1554)&& isprime(p+2800)&& isprime(p+4680) && !isprime(p+7380), print1(p,", ")))

Edited by Wolfdieter Lang, Feb 20 2015

Corrected and extended by Harvey P. Dale, Sep 10 2021

A290530 Primes congruent to (11,17) mod 30.

Original entry on oeis.org

11, 17, 41, 47, 71, 101, 107, 131, 137, 167, 191, 197, 227, 251, 257, 281, 311, 317, 347, 401, 431, 461, 467, 491, 521, 557, 587, 617, 641, 647, 677, 701, 761, 797, 821, 827, 857, 881, 887, 911, 941, 947, 971, 977, 1031, 1061, 1091, 1097, 1151, 1181, 1187, 1217, 1277

Offset: 1

Vincenzo Librandi, Aug 05 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A045372, A187058 (a subsequence).

[p: p in PrimesUpTo(1500) | p mod 30 in [11,17]];

Select[Prime@Range[210], MemberQ[{11, 17}, Mod[#, 30]] &]

A385824 Primes p such that p + 10, p + 18, p + 24, p + 28 and p + 30 are also primes.

Original entry on oeis.org

13, 43, 79, 14533, 41203, 42433, 47119, 88789, 113143, 150193, 340909, 348433, 416389, 556243, 576193, 609589, 626599, 637699, 669649, 715849, 752263, 855709, 859249, 891799, 1107763, 1146763, 1189603, 1191079, 1201999, 1210369, 1225099, 1416043, 1510189, 1601599, 1893163

Offset: 1

Alexander Yutkin, Jul 09 2025

nonn

Initial members of prime sextuples that correspond to the difference pattern [10, 8, 6, 4, 2]. The primes in a sextuple do not have to be consecutive.

			p=13: 13+10=23, 13+18=31, 13+24=37, 13+28=41, 13+30=43 —> prime sextuple: (13, 23, 31, 37, 41, 43).

Cf. A000040.

Cf. A187057 [2, 4, 6, 8], A385035 [8, 6, 4, 2], A187058 [2, 4, 6, 8, 10].

Select[Prime[Range[150000]], And @@ PrimeQ[# + {10, 18, 24, 28, 30}] &] (* Amiram Eldar, Jul 09 2025 *)

cp's OEIS Frontend

A187060 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..7.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A144051 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A190817 Initial primes of 6 consecutive primes with consecutive gaps 2,4,6,8,10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Extensions

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A247949 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A247966 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...6.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A248206 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...7.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A253915 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields primes for k = 0..8, but not for k = 9.

Views