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 1-7 of 7 results.

A115882 Numbers k such that k + prime(k) gives a triangular number.

Original entry on oeis.org

1, 20, 49, 65, 103, 176, 279, 284, 299, 437, 513, 553, 656, 973, 1271, 1779, 1921, 2156, 2312, 2347, 2554, 2759, 3176, 3379, 4008, 4028, 4132, 5255, 6354, 6764, 7116, 8299, 8334, 8366, 8723, 9277, 9755, 10092, 10475, 10631, 11429, 11842, 12633, 13157, 13627
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Examples

			103 + prime(103) = 103 + 563 = 666 = T(36).

Links

Crossrefs

Cf. A064371, A115883, A115886.

Programs

  • Mathematica
    TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; Select[Range[20000], TriangularQ[# + Prime[#]] &] (* T. D. Noe, Jan 27 2014 *)
  • PARI
    isok(n) = ispolygonal(n + prime(n), 3); \\ Michel Marcus, Jan 25 2014

A076147 Numbers k such that the k-th prime + k is a cube.

Original entry on oeis.org

3, 8, 147, 355, 503, 4417, 6288, 22234, 69229, 93068, 105397, 133205, 908767, 1993176, 2081117, 2619491, 2853730, 3559940, 3585297, 3792049, 4228461, 5228796, 6117140, 7624645, 7707795, 9260828, 10435784, 10691791, 11323477
Offset: 1

Views

Author

Zak Seidov, Nov 02 2002

Keywords

Comments

k such that the k-th prime + k is a square gives A064371.

Examples

			3 is a term because 3+prime(3) = 3+5 = 2^3.

Links

Crossrefs

Cf. A064371.

Programs

  • PARI
    isok(n) = ispower(prime(n)+n, 3); \\ Michel Marcus, Oct 02 2013

Extensions

a(9)-a(29) from Donovan Johnson, Jul 02 2010

A086968 Numbers n such that n^2 is a sum k-th prime + k for some k.

Original entry on oeis.org

4, 7, 23, 26, 35, 39, 41, 42, 57, 64, 77, 98, 102, 103, 108, 111, 140, 156, 161, 163, 167, 184, 185, 186, 216, 228, 267, 269, 302, 317, 318, 324, 327, 331, 333, 334
Offset: 1

Views

Author

Zak Seidov, Sep 22 2003

Keywords

Crossrefs

Cf. A064371.

A256246 Numbers n such that 2*n + prime(n) is a square.

Original entry on oeis.org

1, 6, 10, 75, 140, 302, 463, 951, 989, 1219, 1297, 1681, 1776, 1877, 1921, 2379, 2662, 2769, 2828, 3499, 3763, 4810, 4959, 5424, 6156, 6238, 6319, 6829, 7820, 8013, 8108, 9178, 11831, 13570, 13939, 14309, 14677, 17661, 19456, 19894, 21946, 22084, 23148, 23470
Offset: 1

Views

Author

Zak Seidov, Mar 20 2015

Keywords

Examples

			a(1) = 1 because 2*n + prime(n) = 2*1 + 2 = 4 = 2^2,
a(2) = 6 because 2*n + prime(n) = 2*6 + 13 = 25 = 5^2,
a(3) = 10 because 2*n + prime(n) = 2*10 + 29 = 49 = 7^2.

Links

Crossrefs

Cf. A064371, A256247.

Programs

  • Magma
    [n: n in [1..3*10^4] | IsSquare(2*n + NthPrime(n))]; // Vincenzo Librandi, Mar 21 2015
  • Mathematica
    Transpose[Select[Table[{2 n, Prime[n]}, {n, 25000}], IntegerQ[Sqrt[Total[#]]]&]][[1]]/2 (* Vincenzo Librandi, Mar 21 2015 *)
Select[Range[25000],IntegerQ[Sqrt[2*#+Prime[#]]]&] (* Harvey P. Dale, Jun 17 2018 *)
  • PARI
    for(n=1,10^4,if(issquare(2*n+prime(n)),print1(n,", "))) \\ Derek Orr, Mar 22 2015

A104269 Prime numbers p such that primepi(p) + p is a square.

Original entry on oeis.org

11, 37, 443, 571, 1049, 1307, 1451, 1523, 2837, 3593, 5233, 8539, 9257, 9439, 10391, 10987, 17579, 21881, 23321, 23909, 25117, 30557, 30893, 31231, 42239, 47123, 64811, 65789, 83089, 91631, 92219, 95747, 97549, 99971, 101197, 101807, 110603, 114487, 120431
Offset: 1

Views

Author

Zak Seidov, Feb 26 2005

Keywords

Comments

A064371(p) + A000040(A064371(p)) = A086968(p)^2.
p^2 is prime + its index A086968; p + p-th prime is a square A064371.
Equals the prime terms of A073945. - Bill McEachen, Oct 26 2021

Examples

			37 is a term because 37 is 12th prime and 37 + 12 = 49 = 7^2.

Crossrefs

Cf. A000040, A000720, A064371, A086968, A064370, A073945.

Programs

  • Maple
    q:= n-> isprime(n) and issqr(n+numtheory[pi](n)):
select(q, [$0..150000])[];  # Alois P. Heinz, Oct 27 2021
  • Mathematica
    Select[Prime@Range[10^4],IntegerQ@Sqrt[PrimePi@#+#]&] (* Giorgos Kalogeropoulos, Oct 26 2021 *)
  • PARI
    isok(n) = isprime(n) && issquare(n + primepi(n)); \\ Michel Marcus, Oct 05 2013

Formula

a(n) = A086968(n)^2 - pi(a(n)).

Extensions

Definition corrected by Michel Marcus, Oct 05 2013

A114066 n plus the n-th prime gives a fourth power.

Original entry on oeis.org

5, 503, 9229, 53132, 1077614, 5673932, 12335313, 18002978, 89720978, 93036654, 133588773, 167057609, 200368277, 266488037, 270168848, 300988608, 379786284, 693620392, 954150797, 1585905060, 1957585697, 2039039592, 2280501932
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Examples

			5 + prime(5) = 5 + 11 = 16 = 2^4.
12335313 + prime(12335313) = 236421376 = 124^4.

Crossrefs

Cf. A014688, subsequence of A064371.

Programs

  • PARI
    isok(n) = ispower(n+prime(n), 4); \\ Michel Marcus, Jan 08 2014

Extensions

a(11)-a(23) from Donovan Johnson, Jul 02 2010

A239280 Powers of 2 that are sum of prime(k) + k for some k.

Original entry on oeis.org

8, 16, 32, 4096
Offset: 1

Views

Author

Zak Seidov, Mar 14 2014

Keywords

Comments

a(5) > 2^47. - Giovanni Resta, Mar 14 2014
a(5) > 2^62 if it exists. - Chai Wah Wu, Apr 28 2018

Examples

			3 + prime(3) = 8 = 2^3.
5 + prime(5) = 16 = 2^4.
9 + prime(9) = 32 = 2^5.
503 + prime(503) = 4096 = 2^12.

Crossrefs

Cf. A014688, A064371, A073856, A107606.
Showing 1-7 of 7 results.

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

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