cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

Labos Elemer, Dec 12 2003

Keywords

Examples

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

Crossrefs

Cf. A090562, A090563, A090100, A090102.

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Labos Elemer, Dec 29 2003

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Labos Elemer, Dec 29 2003

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

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

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

Views

Author

Labos Elemer, Dec 29 2003

Keywords

Comments

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.

Examples

			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}.

Links

Crossrefs

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

Programs

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

Extensions

Data corrected by Amiram Eldar, Sep 27 2024

A090110 Values of k such that {P(k), P(k+1), ..., P(k+7)} are all prime numbers, where P(k) = 4*k^2 - 154*k + 1523.

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, 66, 129, 130, 328, 1619, 7509, 29714, 45905, 447588, 509862, 1022565, 1102373, 1388125, 1665379, 1762387, 1786292, 2111602, 2962834, 3391838
Offset: 1

Views

Author

Labos Elemer, Dec 30 2003

Keywords

Comments

The terms are arguments introducing a sequence of 8 polynomially consecutive primes with respect to 4*x^2 - 154*x + 1523, a polynomial communicated by Rivera (2003).

Examples

			k = 1 provides {1373, 1231, 1097, 971, 853, 743, 641, 547}, an 8-chain of primes.

Links

Crossrefs

Cf. A090101, A090102, A090107, A090108, A090109, A055561, A090563.

Programs

  • Mathematica
    okQ[x_] := And@@PrimeQ[Table[4n^2-154n+1523, {n,x,x+7}]];
Select[Range[ 510000], okQ] (* Harvey P. Dale, May 25 2011 *)
  • PARI
    isp(x) = isprime(4*x^2 - 154*x + 1523);
lista(kmax) = {my(v = vector(8, k, isp(k))); for(k = 9, kmax, if(vecprod(v) == 1, print1(k - 8, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

Extensions

a(43)-a(51) from Amiram Eldar, Sep 27 2024

A090104 Decimal value of maximal digit of the n-th primorial when written in base n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 20, 21, 22, 23, 24, 25, 24, 27, 27, 29, 29, 31, 32, 33, 32, 35, 35, 35, 34, 39, 39, 41, 40, 43, 44, 44, 45, 45, 48, 49, 50, 51, 52, 53, 54, 53, 56, 57, 58, 59, 60, 61, 61, 62, 64, 64, 66, 67, 68, 69, 70, 69, 72
Offset: 1

Views

Author

Labos Elemer, Dec 17 2003

Keywords

Examples

			n=11: 11th primorial is 200560490130 in base 10 while it is 7806X193790 written in base 11, so a[11]=10 (denoted by X).

Crossrefs

Cf. A090102.

Programs

  • Mathematica
    qp[x_] := Prime[x]*qp[x-1]; {qp[0]=1, qp[1]=2};RecursionLimit$=10000; Table[Max[IntegerDigits[qp[j], j]], {j, 2, 100}]

A090111 Values of k such that {P(k), P(k+1), ..., P(k+6)} are all prime numbers, where P(k) = 4*k^2 - 154*k + 1523.

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, 45, 53, 66, 67, 84, 129, 130, 131, 266, 328, 329, 1619, 1620, 2655, 2937, 7509, 7510, 18030, 29283, 29714, 29715, 37630, 42037, 44473, 45905
Offset: 1

Views

Author

Labos Elemer, Dec 30 2003

Keywords

Comments

The terms are arguments providing a sequence of 7 polynomially consecutive primes with respect to 4*x^2 - 154*x + 1523, a polynomial communicated by Rivera (2003).

Examples

			k = 1 provides {1373, 1231, 1097, 971, 853, 743, 641}, a 7-chain of primes.

Links

Crossrefs

Cf. A090101, A090102, A090107, A090108, A090109, A090110, A055561, A090563.

Programs

  • Mathematica
    Flatten[Position[Partition[Table[If[PrimeQ[4n^2-154n+1523],1,0],{n,46000}],7,1],{1,1,1,1,1,1,1}]] (* Harvey P. Dale, Mar 06 2015 *)
  • PARI
    isp(x) = isprime(4*x^2 - 154*x + 1523);
lista(kmax) = {my(v = vector(7, k, isp(k))); for(k = 8, kmax, if(vecprod(v) == 1, print1(k - 7, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

A090106 Values of k such that {P(k), P(k+1), ..., P(k+12)} are all prime numbers, where P(k) = k^2 + k + 41.

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, 219
Offset: 1

Views

Author

Labos Elemer, Dec 22 2003

Keywords

Comments

a(n) is the first argument providing 13 "polynomially consecutive" primes with respect to the polynomial x^2 + x + 41.
a(29) > 5*10^9, if it exists. - Amiram Eldar, Sep 27 2024

Examples

			k = 219: {P(219), ..., P(231)} = {48221, ..., 53633}, i.e., 13 consecutive integer values substituted to P(x) = x^2 + x + 41 polynomial, all provide primes. The "classical case" includes one single 41-chain of PC-primes, see A055561.

Crossrefs

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

Programs

  • Mathematica
    Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 + k + 41], {k, 1, 1000}], 13, 1], 1] // Flatten (* Amiram Eldar, Sep 27 2024 *)
  • PARI
    isp(x) = isprime(x^2 + x + 41);
lista(kmax) = {my(v = vector(13, k, isp(k))); for(k = 14, kmax, if(vecprod(v) == 1, print1(k - 13, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

Extensions

2 wrong terms removed by Amiram Eldar, Sep 27 2024
Showing 1-8 of 8 results.

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