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.

A243892 a(n) = prime(k) with k = n^2 + prime(n)^2.

Original entry on oeis.org

11, 41, 139, 313, 839, 1259, 2273, 2953, 4493, 7417, 8689, 12659, 15881, 17837, 21683, 28097, 35401, 38321, 46993, 53353, 56909, 67499, 75277, 87539, 105167, 115061, 120431, 130817, 136559, 147881, 189127, 202493
Offset: 1

Views

Author

Freimut Marschner, Jun 14 2014

Keywords

Examples

			n = 1, prime(1^2+prime(1)^2) = prime(1 + 2^2) = prime(5) = 11.
n = 2, prime(2^2+prime(2)^2) = prime(4 + 3^2) = prime(13) = 41.

Links

Crossrefs

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A106587 (n^2 + prime(n)^2).
Also A011757 is prime(n^2), A096327 is prime(prime(n)^2).

Formula

a(n) = prime((n^2 + prime(n)^2)) = prime(A106587(n)).

A243893 a(n) = prime(k-1) with k = n^2 + prime(n)^2.

Original entry on oeis.org

7, 37, 137, 311, 829, 1249, 2269, 2939, 4483, 7411, 8681, 12653, 15877, 17827, 21673, 28087, 35393, 38317, 46957, 53327, 56897, 67493, 75269, 87523, 105143, 115057, 120427, 130811, 136547, 147863, 189067, 202481, 222991, 230393, 267401, 275677
Offset: 1

Views

Author

Freimut Marschner, Jun 14 2014

Keywords

Comments

prime(k-1) is also the largest prime number < (n^2 + prime(n)^2). Remark : Largest prime number < n^2 is A053001. Largest prime number < n^3 is A077037.

Examples

			n=1, 1^2=1, prime(1)^2 = 4, 1 + 4 = 5, 5 - 1= 4, prime(4) = 7 ;
n=2, 2^2=4, prime(2)^2 = 9, 4 + 9= 13, 13 - 1= 12, prime(12) = 37.

Links

Crossrefs

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A106587 (n^2 + prime(n)^2).

Programs

  • Mathematica
    a[n_]:=Prime[(n^2 + Prime[n]^2) - 1]; Array[a,36] (* Stefano Spezia, Mar 12 2025 *)

Formula

a(n) = prime((n^2 + prime(n)^2) - 1) = prime(A106587(n) - 1).

A177150 Numbers k such that k^2 + prime(k)^2 is a prime.

Original entry on oeis.org

1, 2, 10, 20, 30, 34, 36, 50, 60, 100, 108, 110, 112, 114, 122, 130, 156, 188, 192, 198, 204, 208, 210, 216, 230, 234, 240, 246, 250, 260, 286, 290, 294, 300, 330, 332, 338, 342, 360, 388, 390, 392, 410, 416, 440, 460, 468, 484, 492, 502, 532, 542, 556, 570
Offset: 1

Views

Author

Michel Lagneau, May 03 2010

Keywords

Examples

			100 is in the sequence because the 100th prime is 541, and 100^2 + 541^2 = 302681 is prime.

Links

Crossrefs

Cf. A106587.

Programs

  • Magma
    [m:m in [1..600] | IsPrime(m^2+NthPrime(m)^2)]; // Marius A. Burtea, Aug 11 2019
  • Maple
    with(numtheory): nn:= 150: T:=array(1..nn):k:=1:for n from 1 to 1764 do:p:=ithprime(n):if type(p^2+n^2,prime)=true then T[k]:=n:k:=k+1: else fi:od:print(T):
  • Mathematica
    Select[Range[600], PrimeQ[#^2 + Prime[#]^2] &] (* Amiram Eldar, Aug 11 2019 *)

A243769 a(n) = prime(n)^3 mod (n^2 + prime(n)^2).

Original entry on oeis.org

3, 1, 23, 18, 17, 147, 181, 59, 577, 864, 577, 724, 471, 1797, 1595, 1757, 1799, 461, 63, 4246, 2427, 2114, 601, 8215, 9613, 7863, 4279, 1743, 10107, 7652, 14673, 11336, 9671, 3132, 4883, 21177, 19229, 16745, 10683, 6961, 2599, 25966, 30141, 18202, 9415, 37803, 1201, 6538, 48203, 31851, 19757, 11819, 53711, 59088, 51463, 42892, 33339, 10016, 78493, 61693, 36487, 39717
Offset: 1

Views

Author

Freimut Marschner, Jun 10 2014

Keywords

Comments

Remark: The sequence (n^2+prime(n)^2) mod prime(n)^2 is A000290 and the sequence (n^2+prime(n)^2) mod n^2 is A001248.

Examples

			prime(4) = 7, 7^3 = 343, 4^2 + 7^2 = 65, 343 mod 65 = 18.

Links

Crossrefs

Cf. A030078 (prime(n)^3), A106587 (n^2 + prime(n)^2).
Cf. A000290 (n^2), A001248 (prime(n)^2).

Formula

a(n) = A030078(n) mod A106587(n).

A243894 a(n) = prime(k+1) with k = n^2 + prime(n)^2.

Original entry on oeis.org

13, 43, 149, 317, 853, 1277, 2281, 2957, 4507, 7433, 8693, 12671, 15887, 17839, 21701, 28099, 35407, 38327, 46997, 53359, 56911, 67511, 75289, 87541, 105173, 115067, 120473, 130829, 136573, 147919, 189139, 202519, 223009, 230449, 267413, 275711
Offset: 1

Views

Author

Freimut Marschner, Jun 14 2014

Keywords

Comments

The prime numbers prime(k-1) = A243893, prime(k) = A243892 and a(n) = prime(k+1) with k = n^2 + prime(n)^2 are forming a triple of successive prime numbers.

Examples

			n=1, n^2 = 1, prime(1) = 2, 2^2 = 4, 1 + 4 = 5, 5 + 1 = 6, prime(6) = 13 ;
n=2, n^2 = 4, prime(2) = 3, 3^2 = 9, 4 + 9 = 13, 13 + 1 = 14, prime(14) = 43.

Links

Crossrefs

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A106587 (n^2 + prime(n)^2).

Programs

  • Mathematica
    Table[Prime[n^2+Prime[n]^2+1],{n,40}] (* Harvey P. Dale, Dec 31 2015 *)
  • PARI
    vector(40, n, prime(n^2 + prime(n)^2 + 1)) \\ Colin Barker, Jun 14 2014

Formula

a(n) = prime((n^2 + prime(n)^2) + 1) = prime(A106587(n) + 1).
Showing 1-5 of 5 results.

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