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 1-5 of 5 results.

A184935 Primes of the form k^2 + prime(k).

Original entry on oeis.org

3, 7, 23, 83, 181, 239, 563, 1013, 1447, 1607, 2129, 2729, 3167, 3881, 4673, 5849, 6481, 7489, 8563, 9719, 11813, 18713, 21563, 25247, 27197, 29221, 33469, 36467, 47977, 50683, 51599, 56237, 69257, 71389, 75731, 96893, 107119, 115163
Offset: 1

Views

Author

Jonathan Vos Post, Feb 02 2011

Keywords

Comments

Primes in A004232. Sequence A064711 has the values of k.

Examples

			3167 is here because 54^2 + prime(54) = 54^2 + 251 = 3167, which is prime.

Links

Crossrefs

Cf. A000040, A000290, A004232, A064711.

Programs

  • Magma
    [ a: k in [0..10000] | IsPrime(a) where a is k^2 + NthPrime(k) ]; // Vincenzo Librandi, Apr 14 2011
  • Mathematica
    Select[Table[k^2 + Prime[k], {k, 1000}], PrimeQ] (* Harvey P. Dale, Feb 16 2011 *)

Extensions

Better name from Zak Seidov, Apr 12 2011

A188831 Primes of the form k^2 - prime(k).

Original entry on oeis.org

23, 71, 107, 263, 487, 677, 787, 1427, 1583, 2081, 3319, 5393, 8713, 10247, 11071, 12377, 18257, 20477, 24659, 26573, 29243, 29927, 33487, 34949, 37223, 37991, 41981, 51449, 60917, 64937, 66977, 71167, 83357, 85667, 99013, 100271, 109313, 110629, 118757
Offset: 1

Views

Author

Zak Seidov, Apr 11 2011

Keywords

Comments

Or, primes in A073497. Corresponding values of k in A064712.
This is to A073497 and A064712 as A184935 is to A004232 and A064711.
The two primes prime(k) and k^2-prime(k) are a Goldbach partition of k^2. - T. D. Noe, Apr 14 2011

Examples

			23 is here because 6^2 - prime(6) = 36 - 13 = 23.

Links

Crossrefs

Cf. A004232, A064711, A064712, A073497, A184935.

Programs

  • Magma
    [ a: k in [0..10000] | IsPrime(a) where a is k^2-NthPrime(k) ]; // Vincenzo Librandi, Apr 14 2011
  • Mathematica
    Select[Table[k^2 - Prime[k], {k, 1000}], PrimeQ] (* T. D. Noe, Apr 14 2011 *)

Formula

a(n) = A073497(A064712(n)).

A064483 Numbers k such that k^2 + prime(k) and k^2 - prime(k) are both primes.

Original entry on oeis.org

12, 30, 60, 96, 336, 660, 702, 756, 984, 990, 1188, 1302, 1488, 1830, 1866, 2070, 2142, 2340, 2586, 2874, 2910, 3618, 3714, 3750, 3774, 3906, 4008, 4470, 4512, 4902, 5094, 5754, 6012, 6174, 6432, 6840, 6846, 6930, 7446, 7578, 7734, 8064, 8190, 8328
Offset: 1

Views

Author

Robert G. Wilson v and Jason Earls, Oct 05 2001

Keywords

Comments

All terms are multiples of 6. - Jon E. Schoenfield, Apr 13 2024

Examples

			12 is in the sequence because 144 +/- 37 = 181 and 107 which are both primes.
k=30 is a term: 30^2 = 900, prime(30) = 113, 900+113 = 1013 and 900-113 = 787, both primes.

Links

Crossrefs

Intersection of A064711 and A064712. - Zak Seidov, Oct 12 2014

Programs

  • Mathematica
    Select[ Range[10^4], PrimeQ[ #^2 + Prime[ # ]] && PrimeQ[ #^2 - Prime[ # ]] &]
  • PARI
    for(n=1,20000, if(isprime(n^2+prime(n)) && isprime(n^2-prime(n)), print1(n," ")))
  • PARI
    { n=0; default(primelimit, 6100000); for (m=1, 10^9, if (isprime(m^2 + prime(m)) && isprime(m^2 - prime(m)), write("b064483.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 16 2009

A228828 Numbers n such that n^2 + pi(n) is prime.

Original entry on oeis.org

2, 3, 7, 12, 18, 21, 36, 37, 42, 45, 52, 55, 60, 61, 65, 68, 70, 79, 84, 95, 98, 113, 130, 135, 143, 145, 155, 180, 181, 185, 195, 205, 216, 222, 231, 239, 253, 262, 273, 275, 325, 332, 334, 354, 368, 370, 385, 402, 417, 421, 432, 433, 454, 462, 488, 505, 516
Offset: 1

Views

Author

K. D. Bajpai, Sep 04 2013

Keywords

Comments

Conjecture: the sequence is infinite.

Examples

			a(6) = 21 :  n^2+pi(n ) = 21^2 + pi(21) = 441+8 = 449 which is a prime.

Links

Crossrefs

Cf. A064711, A064712.
Cf. A077510 (numbers n such that n + pi(n) is a prime).

Programs

  • Maple
    with(numtheory): KD:= proc() local a;  a:= n^2+pi(n); if isprime(a) then RETURN(n): fi; end: seq(KD(), n=1..2000);
  • Mathematica
    Select[Range[600],PrimeQ[#^2+PrimePi[#]]&] (* Harvey P. Dale, Jul 04 2018 *)
  • PARI
    v=List(); p=0; for(n=2,1e4,p+=isprime(n); if(isprime(n^2+p), listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Sep 04 2013

A253971 Prime(n) is included iff prime(n) + n^2 is also prime.

Original entry on oeis.org

2, 3, 7, 19, 37, 43, 79, 113, 151, 163, 193, 229, 251, 281, 317, 373, 397, 433, 463, 503, 577, 757, 827, 911, 953, 997, 1069, 1123, 1321, 1399, 1423, 1481, 1657, 1693, 1747, 2029, 2143, 2267, 2311, 2473, 2503, 2551, 2593, 2687, 2753, 2791, 2917, 3313, 3323
Offset: 1

Views

Author

Vincenzo Librandi, Feb 04 2015

Keywords

Examples

			7 is in this sequence because 7+16=23.
19 is in this sequence because 19+64=83.

Crossrefs

Cf. A064711, A061067.

Programs

  • Magma
    [NthPrime(n): n in [1..500] | IsPrime(NthPrime(n)+n^2)];
  • Mathematica
    Prime[Select[Range[500], PrimeQ[Prime[#] + #^2] &]]
  • PARI
    lista(nn) = forprime (n=2, nn, if (isprime(n+primepi(n)^2), print1(n, ", "))); \\ Michel Marcus, Feb 04 2015

Formula

a(n) = prime(A064711(n)). - Michel Marcus, Feb 04 2015
Showing 1-5 of 5 results.

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