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 1-6 of 6 results.

A229080 Primes of the form T(k) + S(k) + 1 where T(k) is the k-th triangular number and S(k) is the k-th square number.

Original entry on oeis.org

3, 41, 101, 127, 223, 443, 673, 877, 1277, 1553, 2543, 2927, 3061, 4241, 4733, 4903, 5431, 6971, 7177, 8933, 9883, 10627, 11927, 12743, 13873, 17551, 19211, 20593, 21661, 26203, 31177, 33377, 36583, 40427, 51431, 58313, 60101, 60703, 67523, 68161, 75377, 81551
Offset: 1

Views

Author

K. D. Bajpai, Sep 12 2013

Keywords

Comments

Also primes of the form (3k^2 + k + 2)/2.

Examples

			T(8) + S(8) + 1 = 8*(8+1)/2 + 8^2 + 1 = 101, which is prime, so 101 is a term.
T(9) + S(9) + 1 = 9*(9+1)/2 + 9^2 + 1 = 127, which is prime, so 127 is a term.

Links

Crossrefs

Cf. A000217, A000290, A120479.

Programs

  • Maple
    KD:= proc() local a,b,d;  a:= (1/2)*x*(x+1); b:=x^2; d:=a+b+1;  if isprime(d) then   RETURN(d): fi; end: seq(KD(),x=1..500);

A229157 Primes of the form T(n) + C(n) - 1 where T(n) and C(n) are n-th triangular and cube numbers.

Original entry on oeis.org

73, 139, 547, 773, 2287, 4231, 8209, 9491, 36497, 111767, 118873, 177211, 217829, 316777, 391717, 596273, 889391, 1005049, 1035451, 1163189, 1265597, 1301023, 1449337, 1735259, 2105407, 2524771, 3319123, 4755827, 5467351, 6246523, 6348829, 6662437, 8636239, 9151273
Offset: 1

Views

Author

K. D. Bajpai, Sep 15 2013

Keywords

Comments

Also primes of the form (2*n^3 + n^2 + n - 2)/2.

Examples

			a(3)=547: T(8)+C(8)-1 = (1/2)*8*(8+1)+8^3-1 = 547 which is prime.
a(4)=773: T(9)+C(9)-1 = (1/2)*9*(9+1)+9^3-1 = 773 which is prime.

Links

Crossrefs

Cf. A120479, A229080.

Programs

  • Maple
    KD:= proc() local a,b,d; a:= (1/2)*n*(n+1);b:=n^3; d:=a+b-1; if isprime(d) then   RETURN(d): fi;end:seq(KD(),n=1..500);

A119264 Primes of the form k^6 + k^2 + 1.

Original entry on oeis.org

3, 739, 2986129, 11390851, 191103553, 387421219, 1291469059, 2176783633, 8303767651, 24794914213, 34296450499, 646990192099, 782757798913, 2194972636933, 3462826006819, 14412774469393, 27752076894853
Offset: 1

Views

Author

Jonathan Vos Post, Jul 23 2006

Keywords

Comments

The numbers k such that k^6 + k^2 + 1 is prime begin 1, 3, 12, 15, 24, 27, 33, 36, 45, 54, 57, 93, 96, 114, 123, 156, 174, 177, 210, ...; other than 1, each is divisible by 3.

Examples

			a(1) = 1^6 + 1^2 + 1 = 3 is prime.
a(2) = 3^6 + 3^2 + 1 = 739 is prime.
a(3) = 12^6 + 12^2 + 1 = 2986129 is prime.
a(4) = 15^6 + 15^2 + 1 = 11390851 is prime.

Crossrefs

Cf. A000040, A120479, A120480.

Programs

  • Magma
    [a: n in [0..250]|IsPrime(a) where a is n^6+n^2+1] // Vincenzo Librandi, Dec 22 2010
  • Mathematica
    Select[Table[n^6+n^2+1,{n,200}],PrimeQ] (* Harvey P. Dale, Oct 17 2019 *)

Formula

a(n) = A000040 INTERSECTION {n^6 + n^2 + 1}.

Extensions

a(1) corrected by Vincenzo Librandi, Dec 22 2010

A119448 Primes of the form k^k + k^2 + 1.

Original entry on oeis.org

2, 3, 37, 285311670733, 1102507499354148695951786433413508348166942596435548101, 3877924263464448622666648186154330754898344901344205917642325627886496385065073
Offset: 1

Views

Author

Jonathan Vos Post, Jul 26 2006

Keywords

Comments

The values of k such that k^k + k^2 + 1 is prime are 1, 3, 11, 35, 47, ...
0^0 + 0^2 + 1 = 2. - Vincenzo Librandi, Dec 22 2010
The next term has 176 digits. - Harvey P. Dale, Feb 07 2018

Examples

			a(3) = 11^11 + 11^2 + 1 = 285311670733 is prime.

Crossrefs

Cf. A000040, A000312, A120479, A120480, A119264.

Programs

  • Magma
    [a: n in [0..250]|IsPrime(a) where a is n^n+n^2+1] // Vincenzo Librandi, Dec 22 2010
  • Mathematica
    Join[{2},Select[Table[n^n+n^2+1,{n,50}],PrimeQ]] (* Harvey P. Dale, Feb 07 2018 *)

Extensions

2 added by Vincenzo Librandi, Dec 22 2010

A228908 Primes of the form T(n) + S(n) + C(n) + 1 where T(n), S(n) and C(n) are the n-th triangular, square and cube numbers.

Original entry on oeis.org

43, 421, 613, 1951, 7411, 12973, 23143, 31249, 48619, 114073, 210631, 256033, 321403, 365509, 381061, 502441, 521641, 669901, 766039, 791431, 1015051, 1108693, 1242271, 1929751, 2121793, 2773471, 3759991, 3832999, 4057681, 5498329, 7133281, 7472011, 7587259
Offset: 1

Views

Author

K. D. Bajpai, Sep 14 2013

Keywords

Comments

Also primes of the form n^3 + 3/2*n^2 + 1/2*n + 1.

Examples

			a(3) = 613: T(8)+S(8)+C(8)+1 = 1/2*8*(8+1)+8^2+8^3+1 = 613 which is prime.
a(4) = 1951: T(12)+S(12)+C(12)+1 = 1/2*12*(12+1)+12^2+12^3+1 = 1951 which is prime.

Links

Crossrefs

Cf. A120479, A229080.

Programs

  • Maple
    KD:= proc() local a,b,c,d; a:= (1/2)*n*(n+1); b:=n^2; c:=n^3; d:=a+b+c+1; if isprime(d) then   RETURN(d): fi; end:seq(KD(),n=1..500);
  • PARI
    select(isprime, vector(100,n,n^3+3/2*n^2+n/2+1)) \\ Charles R Greathouse IV, Sep 15 2013

A119987 Primes of the form k^k + k^3 + 1.

Original entry on oeis.org

2, 3, 13, 3251, 16777729, 387421219, 11112006825560761, 443426488243037769948249630619149912487, 1075911801979993982060429252856123779115487368830416064665177
Offset: 1

Views

Author

Jonathan Vos Post, Aug 03 2006

Keywords

Comments

The values of k such that k^k + k^2 + 1 is prime are 0, 1, 2, 5, 8, 9, 14, 27, 38, 62, 255, (4000), .... per Vincenzo Librandi's extension, 2 is an element iff one defines 0^0 = 1.

Links

Crossrefs

Cf. A000040, A120479, A120480, A119264, A119448.

Programs

  • Magma
    [ a: n in [0..250] | IsPrime(a) where a is n^n+n^3+1 ]; // Vincenzo Librandi, Dec 22 2010
  • Mathematica
    f[n_] := (n^n + n^3 + 1); Select[f@ Range@ 40, PrimeQ]
  • PARI
    for(n=1, 255, if(ispseudoprime(t=n^n+n^3+1), print1(t", "))); v \\ Charles R Greathouse IV, Feb 17 2011

Extensions

2 added by Vincenzo Librandi, Dec 22 2010
Showing 1-6 of 6 results.

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