A090562 -id:A090562 - cp's OEIS frontend

A090100 Numbers n such that n and the four successive integers produce primes if substituted for x in the polynomial 5x^2+5x+1. See A090562, A090563. Terms show that longer similar chains also exist.

1, 2, 3, 13, 266, 321, 322, 323, 344, 641, 1324, 5436, 16700, 16701, 19857, 19858, 28151, 28152, 30648, 31253, 32045, 45773, 48710, 50923, 52397, 57357, 57358, 63879, 63880, 63881, 72615, 73164, 73165, 78785, 81831, 87640, 87641, 91116

Offset: 1

Labos Elemer, Dec 12 2003

For examples of longer similar chains, if n = 1, 321, or 63879, the polynomial produces 7 consecutive prime terms (including n). - Harvey P. Dale, May 04 2024

Cf. A090562, A090563.

Do[s=5*n^2+5*n+1;s1=5*(n+1)^2+5*(n+1)+1; s2=5*(n+2)^2+5*(n+2)+1;s3=5*(n+3)^2+5*(n+3)+1; s4=5*(n+4)^2+5*(n+4)+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&& PrimeQ[s3]&&PrimeQ[s4], Print[n]], {n, 1, 100000}]
SequencePosition[Table[If[PrimeQ[5n^2+5n+1],1,0],{n,100000}],{1,1,1,1,1}][[;;,1]] (* Harvey P. Dale, May 04 2024 *)

A090563 Numbers k such that 5k^2 + 5k + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 11, 13, 14, 15, 16, 17, 19, 21, 22, 24, 25, 26, 30, 32, 35, 36, 38, 43, 44, 49, 52, 54, 55, 60, 61, 62, 63, 66, 69, 70, 71, 73, 74, 79, 80, 81, 82, 85, 87, 88, 93, 94, 101, 104, 106, 107, 109, 110, 112, 114, 115, 117, 123, 127, 128, 131, 135

Offset: 1

Amarnath Murthy, Dec 11 2003

nonn

Robert Price, Table of n, a(n) for n = 1..2468

Cf. A090562.

Do[s=5*n^2+5*n+1; If[PrimeQ[s], Print[n]], {n, 1, 1000}]
Select[Range[1000], PrimeQ[5 #^2 + 5 # + 1] &] (* Robert Price, Apr 13 2025 *)

isok(n) = isprime(5*n^2+5*n+1); \\ Michel Marcus, Apr 30 2016

More terms from Labos Elemer, Dec 12 2003

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A090102 Leading prime in each set of 7 arising in A090101.

Original entry on oeis.org

11, 516811, 20402952601, 196260616589761, 239536538008051, 426813020692661, 2681027962124411, 3605832801512401, 6450361508166761, 10392841156929031, 13162202092936411, 13655671002023851, 14501847401205811

Offset: 1

Labos Elemer, Dec 15 2003

nonn

			a[15] = 69981018761651281 is first of following chain: {69981018761651281, 69981019944706811, 69981021127762351, 69981022310817901, 69981023493873461, 69981024676929031, 69981025859984611} = {P[k], P[k+1], ..., P[k+6]}, where k = A090101[15] and P[x] = 5x^2+5x+1. See A090562, A090563.

Cf. A090562, A090563, A090100, A090101, A056561.

po[x_] := 5*x^2+5*x+1 Do[s=po[n];s0=po[n];s1=po[n+1];s2=po[n+2];s3=po[n+3];s4=po[n+4]; s5=po[n+5];s6=po[n+6];If[IntegerQ[n/100000], Print[{n}]]; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5] &&PrimeQ[s6], Print[s0]], {n, 1, 120000000}]

A090101 Numbers n such that n and the 6 successive integers yield primes if substituted for x in polynomial 5x^2+5x+1.

Original entry on oeis.org

1, 321, 63879, 6265151, 6921510, 9239188, 23156113, 26854544, 35917576, 45591317, 51307313, 52260254, 53855078, 71731838, 118305552, 124220571, 124234464, 150767861, 170448863, 192850264

Offset: 1

Labos Elemer, Dec 12 2003

nonn

			a[15]=118305552 and the corresponding seven "polynomially consecutive" primes are: {69981018761651281, 69981019944706811, 69981021127762351, 69981022310817901, 69981023493873461, 69981024676929031, 69981025859984611}

Cf. A090562, A090563, A090100, A090102.

po[x_] := 5*x^2+5*x+1 Do[s=po[n];s0=po[n];s1=po[n+1];s2=po[n+2];s3=po[n+3];s4=po[n+4]; s5=po[n+5];s6=po[n+6];If[IntegerQ[n/100000], Print[{n}]]; If[PrimeQ[s0]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5] &&PrimeQ[s6], Print[{n, s0, s1, s2, s3, s4, s5, s6}]], {n, 1, 120000000}]
Select[Range[193*10^6],AllTrue[Table[5x^2+5x+1,{x,Range[#,#+6]}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 02 2020 *)

More terms from Don Reble, Dec 14 2003

A090107 Values of k such that {P(k), P(k+1), ..., P(k+9)} are all prime numbers, where P(k) = k^2 - 79*k + 1601.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 106

Offset: 1

Labos Elemer, Dec 29 2003

nonn

a(n) is the first argument providing 10 "polynomially consecutive" primes with respect to the polynomial x^2 - 79*x + 1601 described by Escott in 1899.

			k = 106 provides a chain of 10 "polynomially consecutive" primes as follows: {4463, 4597, 4733, 4871, 5011, 5153, 5297, 5443, 5591, 5741}.

Amiram Eldar, Table of n, a(n) for n = 1..145
E. B. Escott, Réponse 1133, Formule d'Euler x^2 + x + 41 et formules analogues, L'Intermédiaire des mathématiciens, Vol. 6 (1899), pp. 10-11.

Cf. A055561, A090101, A090102, A090562, A090563.

Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 - 79*k + 1601], {k, 1, 1000}], 10, 1], 1] // Flatten (* Amiram Eldar, Sep 27 2024 *)

isp(x) = isprime(x^2 - 79*x + 1601);
lista(kmax) = {my(v = vector(10, k, isp(k))); for(k = 11, kmax, if(vecprod(v) == 1, print1(k - 10, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

Data corrected by Amiram Eldar, Sep 27 2024

A090108 Values of k such that {P(k), P(k+1), ..., P(k+8)} are all prime numbers, whereP(k) = k^2 - 79*k + 1601.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 106

Offset: 1

Labos Elemer, Dec 29 2003

nonn

a(n) is the first argument providing 9 "polynomially consecutive" primes with respect to the polynomial x^2 - 79*x + 1601 described by Escott in 1899.

			k = 263 provides a chain of 9 "polynomially consecutive" primes as follows: {49993, 50441, 50891, 51343, 51797, 52253, 52711, 53171, 53633}.

Amiram Eldar, Table of n, a(n) for n = 1..535
E. B. Escott, Réponse 1133, Formule d'Euler x^2 + x + 41 et formules analogues, L'Intermédiaire des mathématiciens, Vol. 6 (1899), pp. 10-11.

Cf. A055561, A090101, A090102, A090107, A090562, A090563.

Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 - 79*k + 1601], {k, 1, 1000}], 9, 1], 1] // Flatten (* Amiram Eldar, Sep 27 2024 *)

isp(x) = isprime(x^2 - 79*x + 1601);
lista(kmax) = {my(v = vector(9, k, isp(k))); for(k = 10, kmax, if(vecprod(v) == 1, print1(k - 9, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

A276261 Centered 21-gonal primes.

Original entry on oeis.org

127, 211, 757, 2521, 2857, 6301, 8527, 16381, 19867, 23689, 24697, 27847, 32341, 37171, 38431, 42337, 66361, 68041, 82237, 89839, 97777, 103951, 114661, 140071, 152461, 162751, 170689, 192781, 204331, 216217, 231547, 240997, 284131, 308827, 353557, 357421, 385057, 389089

Offset: 1

Ilya Gutkovskiy, Aug 26 2016

nonn

Primes of the form (21*k^2 + 21*k + 2)/2.

Numbers k such that (21*k^2 + 21*k + 2)/2 is prime: 3, 4, 8, 15, 16, 24, 28, 39, 43, 47, 48, 51, 55, 059, 60, 63, 79, 80, 88, 92, 96, 99, ...

OEIS Wiki, Centered polygonal numbers
Eric Weisstein's World of Mathematics, Centered Polygonal Number
Index entries for sequences related to centered polygonal numbers

Cf. A000040, A069178.

Cf. similar sequences of the centered k-gonal primes: A125602 (k = 3), A027862 (k = 4), A145838 (k = 5), A002407 (k = 6), A144974 (k = 7), A090562 (k = 10), A262344 (k = 11), A262493 (k = 13), A264821 (k = 14), A264822 (k = 15), A264823 (k = 16), A264824 (k = 17), A264825 (k = 18), A264844 (k = 19), A264845 (k = 20), A201715 (k = 24).

Intersection[Table[(21 k^2 + 21 k + 2)/2, {k, 0, 1000}], Prime[Range[33000]]]

lista(nn) = for(n=1, nn, if(isprime(p=(21*n^2 + 21*n + 2)/2), print1(p, ", "))); \\ Altug Alkan, Aug 26 2016

A090109 Values of k such that {P(k), P(k+1), ..., P(k+10)} are all prime numbers, where P(k) = k^2 - 79*k + 1601.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 259, 260, 261

Offset: 1

Labos Elemer, Dec 29 2003

nonn

a(n) is the first argument providing 11 "polynomially consecutive" primes with respect to the polynomial x^2 - 79*x + 1601 described by Escott in 1899.

			k = 1 provides the following non-monotonic (!) chain of 11 "polynomially consecutive" primes as follows: {1523, 1447, 1373, 1301, 1231, 1163, 1097, 1033, 971, 911, 853}.

Amiram Eldar, Table of n, a(n) for n = 1..83
E. B. Escott, Réponse 1133, Formule d'Euler x^2 + x + 41 et formules analogues, L'Intermédiaire des mathématiciens, Vol. 6 (1899), pp. 10-11.

Cf. A055561, A090101, A090102, A090107, A090108, A090562, A090563.

Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 - 79*k + 1601], {k, 1, 1000}], 11, 1], 1] // Flatten (* Amiram Eldar, Sep 27 2024 *)

isp(x) = isprime(x^2 - 79*x + 1601);
lista(kmax) = {my(v = vector(11, k, isp(k))); for(k = 12, kmax, if(vecprod(v) == 1, print1(k - 11, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

Data corrected by Amiram Eldar, Sep 27 2024

A134462 Centered decagonal palindromic primes; or palindromic primes of the form 5k^2 + 5k + 1.

Original entry on oeis.org

11, 101, 151, 1598951, 1128512158211, 104216919612401, 107635959536701, 106906347292743609601, 165901968762984246868642489267869109561

Offset: 1

Alexander Adamchuk, Oct 26 2007

Sequence is the intersection of the palindromic primes = A002385 = {2, 3, 5, 7, 11, 101, 131, 151, ...} and the centered 10-gonal numbers = A062786 = {1, 11, 31, 61, 101, 151, ...}. Corresponding numbers k such that 5k^2 + 5k + 1 is a term of A134462 are listed in A134463 = {1, 4, 5, 565, 475081, ...}. Note that the first 4 terms of A134463 are palindromic as well.
a(9) > 10^25. - Donovan Johnson, Feb 13 2011
a(10) > 10^39. - Patrick De Geest, May 29 2021

Patrick De Geest, Palindromic Centered Polygonal Numbers
Eric Weisstein's World of Mathematics, Palindromic Prime
Wikipedia, Centered decagonal number.

Cf. A002385 = Palindromic primes.
Cf. A062786 = Centered 10-gonal numbers.
Cf. A090562 = Primes of the form 5k^2 + 5k + 1.
Cf. A090563 = Values of k such that 5k^2 + 5k + 1 is a prime.
Cf. A134463 = Values of k such that 5k^2 + 5k + 1 is a palindromic prime.

Do[ f=5k^2+5k+1; If[ PrimeQ[f] && FromDigits[ Reverse[ IntegerDigits[ f ] ] ] == f, Print[ f ] ], {k, 1, 500000} ]

More terms from Tomas J. Bulka (tbulka(AT)rodincoil.com), Aug 30 2009
a(8) from Donovan Johnson, Feb 13 2011
a(9) from Patrick De Geest, May 29 2021

A134463 Values of k such that 5k^2 + 5k + 1 is a palindromic prime.

Original entry on oeis.org

1, 4, 5, 565, 475081, 4565455, 4639740, 4623988479, 5760242508141202328

Offset: 1

Alexander Adamchuk, Oct 26 2007

Corresponding centered decagonal palindromic primes are 5k^2 + 5k + 1 = A134462 = {11, 101, 151, 1598951, 1128512158211, ...}. Note that the first 4 terms of A134463 are palindromic as well.
a(9) > 1414213562372. - Donovan Johnson, Feb 13 2011
a(10) > 14142135623730950488. - Patrick De Geest, May 29 2021

Patrick De Geest, Palindromic Centered Polygonal Numbers
Eric Weisstein's World of Mathematics, Palindromic Prime
Wikipedia, Centered decagonal number

Cf. A134462 = Centered decagonal palindromic primes; or palindromic primes of the form 5k^2 + 5k + 1.
Cf. A002385 = Palindromic primes.
Cf. A062786 = Centered 10-gonal numbers.
Cf. A090562 = Primes of the form 5k^2 + 5k + 1.
Cf. A090563 = Values of k such that 5k^2 + 5k + 1 is a prime.

Do[ f=5k^2+5k+1; If[ PrimeQ[f] && FromDigits[ Reverse[ IntegerDigits[ f ] ] ] == f, Print[ k ] ], {k, 1, 500000} ]

a(6), a(7) from D. S. McNeil, Mar 02 2009
a(8) from Donovan Johnson, Feb 13 2011
a(9) from Patrick De Geest, May 29 2021

cp's OEIS Frontend

A090100 Numbers n such that n and the four successive integers produce primes if substituted for x in the polynomial 5x^2+5x+1. See A090562, A090563. Terms show that longer similar chains also exist.

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A090563 Numbers k such that 5*k^2 + 5*k + 1 is prime.

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A090102 Leading prime in each set of 7 arising in A090101.

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A090101 Numbers n such that n and the 6 successive integers yield primes if substituted for x in polynomial 5x^2+5x+1.

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A090107 Values of k such that {P(k), P(k+1), ..., P(k+9)} are all prime numbers, where P(k) = k^2 - 79*k + 1601.

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A090108 Values of k such that {P(k), P(k+1), ..., P(k+8)} are all prime numbers, whereP(k) = k^2 - 79*k + 1601.

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A276261 Centered 21-gonal primes.

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A090109 Values of k such that {P(k), P(k+1), ..., P(k+10)} are all prime numbers, where P(k) = k^2 - 79*k + 1601.

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A090563 Numbers k such that 5k^2 + 5k + 1 is prime.