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.

Previous Showing 11-15 of 15 results.

A125239 Smallest prime divisor of 10*T(n)+1 = 5*n*(n+1)+1, where T(n) = 1 + 2 + ... + n.

Original entry on oeis.org

11, 31, 61, 101, 151, 211, 281, 19, 11, 19, 661, 11, 911, 1051, 1201, 1361, 1531, 29, 1901, 11, 2311, 2531, 11, 3001, 3251, 3511, 19, 31, 19, 4651, 11, 5281, 31, 11, 6301, 6661, 79, 7411, 29, 59, 79, 11, 9461, 9901, 11, 19, 29, 19, 12251, 41, 89, 13781, 11
Offset: 1

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Author

Nick Hobson, Nov 25 2006

Keywords

Comments

All divisors of 10*T(n)+1 are congruent to 1 or -1 modulo 10; that is, they end in the decimal digit 1 or 9.

Examples

			10*T(9) + 1 = 5*9*10 + 1 = 451 = 11*41, so a(9) = 11.

Links

Crossrefs

Cf. A000217, A062786, A090562, A124989.

Programs

  • Mathematica
    FactorInteger[#][[1,1]]&/@(10*Accumulate[Range[60]]+1) (* Harvey P. Dale, Dec 12 2011 *)
  • PARI
    a(n) = if(n<1, 0, factor(5*n*(n+1)+1)[1,1])

A201786 Primes of the form 5*k^2 - 4.

Original entry on oeis.org

41, 241, 401, 601, 1801, 3121, 4201, 4801, 5441, 6121, 6841, 9241, 13001, 15121, 17401, 19841, 21121, 22441, 23801, 26641, 29641, 32801, 45121, 47041, 51001, 57241, 63841, 75641, 78121, 91121, 96601, 99401, 102241, 108041, 114001, 117041
Offset: 1

Views

Author

Vincenzo Librandi, Dec 05 2011

Keywords

Comments

Also, primes of the form 20*k^2 + 20*k + 1. - Jan Rider, May 22 2018

Links

Crossrefs

Cf. A090562, A195317.

Programs

  • Magma
    [a: n in [1..400] | IsPrime(a) where a is 5*n^2-4];
  • Mathematica
    Select[Table[5n^2-4,{n,1,1000}],PrimeQ]
  • PARI
    lista(nn) = for (n=1, nn, if (isprime(p=5*n^2-4), print1(p, ", "));); \\ Michel Marcus, May 22 2018

A320752 Primes of the form 5*n^2 - 5*n + 13.

Original entry on oeis.org

13, 23, 43, 73, 113, 163, 223, 293, 373, 463, 563, 673, 1063, 1213, 1373, 1543, 1723, 1913, 2113, 2543, 3793, 4073, 4363, 4663, 4973, 5623, 6673, 7043, 8623, 9043, 9473, 12263, 12763, 14323, 15413, 15973, 17123, 17713, 18313, 19543, 20173, 22123, 23473, 26293
Offset: 1

Views

Author

Arashdeep Singh, Oct 20 2018

Keywords

Comments

The first 12 numbers of the form 5*n^2 - 5*n + 13 (n=1 to 12) are primes.

Links

Crossrefs

Cf. A090562.

Programs

  • GAP
    Filtered(List([1..75],n->5*n^2-5*n+13),IsPrime); # Muniru A Asiru, Oct 21 2018
  • Maple
    select(isprime,[seq(5*n^2-5*n+13,n=1..75)]); # Muniru A Asiru, Oct 21 2018
  • Mathematica
    Select[Table[5n^2-5n+13,{n,80}],PrimeQ] (* Harvey P. Dale, Aug 22 2021 *)
  • PARI
    terms(n) = my(i=0); for(k=1, oo, my(x=5*k^2-5*k+13); if(ispseudoprime(x), print1(x, ", "); i++); if(i==n, break))
/* Print initial 50 terms as follows */
terms(50) \\ Felix Fröhlich, Oct 20 2018

Extensions

More terms from Felix Fröhlich, Oct 20 2018

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

A298760 Numbers k such that there is a record number of consecutive prime centered k-gonal numbers after 1.

Original entry on oeis.org

1, 2, 6, 10, 46, 102, 7186, 6382932
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2018

Keywords

Comments

The number of consecutive primes is 1, 3, 4, 7, 8, 9, 10, 11.

Examples

			The first 8 centered 10-gonal numbers (A062786) are 1, 11, 31, 61, 101, 151, 211, 281, and all of them except for 1 are primes (A090562). The previous record is 4 primes, for centered hexagonal numbers 7, 19, 37, 61 (A003215), therefore 6 and 10 are in the sequence.
From _Michel Marcus_, Feb 12 2018: (Start)
  Number of primes after the 1
1: 1  2  4  7  11  16 ...  : 1   <- record
2: 1  3  7 13  21  31 ...  : 3   <- record
3: 1  4 10 19  31  46 ...  : 0
4: 1  5 13 25  41  61 ...  : 2
5: 1  6 16 31  51  76 ...  : 0
6: 1  7 19 37  61  91 ...  : 4   <- record
....
(End)

Links

Crossrefs

Cf. A000124, A002061, A003215, A062786, A090562.

Programs

  • Mathematica
    f[n_, k_] := k*n (n - 1)/2 + 1; a[k_] := Module[{n = 2}, While[PrimeQ[f[n, k]], n++]; n - 2]; am = 0; seq={}; Do[a1 = a[n]; If[a1 > am, AppendTo[seq, n]; am = a1], {n,1,10^7}]; seq
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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