A243451 -id:A243451 - cp's OEIS frontend

A256839 Primes of form n^2 + 14641.

14657, 14741, 14897, 15217, 15541, 15797, 15937, 19541, 20117, 22037, 22741, 23857, 25457, 28097, 30517, 31541, 38977, 40241, 42197, 43541, 44917, 47041, 48497, 50741, 57077, 58741, 61297, 64817, 65717, 74177, 77141, 80177, 82241, 87541, 107057, 117041

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256840 (b=12), A256841 (b=13).

a256839 n = a256839_list !! (n-1)
a256839_list = [x | x <- map (+ 14641) a000290_list, a010051' x == 1]

Select[Range[500]^2+14641,PrimeQ] (* Harvey P. Dale, Mar 20 2017 *)

A256840 Primes of form n^2 + 20736.

Original entry on oeis.org

20857, 21577, 21961, 23761, 27961, 28657, 29017, 29761, 30937, 33961, 34897, 37897, 41761, 42937, 49297, 51361, 60337, 62761, 65257, 80761, 83737, 93097, 107761, 111337, 113761, 122497, 132961, 142537, 151057, 164377, 173617, 181537, 188017, 192961, 218761

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256841 (b=13).

a256840 n = a256840_list !! (n-1)
a256840_list = [x | x <- map (+ 20736) a000290_list, a010051' x == 1]

A256841 Primes of form n^2 + 28561.

Original entry on oeis.org

28597, 28661, 28817, 28961, 29137, 29717, 30161, 30497, 30677, 31477, 32917, 33461, 34337, 34961, 35617, 37397, 38561, 42017, 42961, 47057, 49297, 49877, 51061, 55457, 60961, 62417, 64661, 66977, 70177, 70997, 72661, 74357, 75217, 76961, 78737, 86161, 93077

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Conjecture: sequence is infinite.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A010051, A000290; subsequence of A028916.

Primes of form n^2+b^4, b fixed: A002496 (b=1), A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12).

a256841 n = a256841_list !! (n-1)
a256841_list = [x | x <- map (+ 28561) a000290_list, a010051' x == 1]

Select[Range[300]^2+28561,PrimeQ] (* Harvey P. Dale, Oct 18 2021 *)

A241751 a(n) = n^2 + 16.

Original entry on oeis.org

16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 116, 137, 160, 185, 212, 241, 272, 305, 340, 377, 416, 457, 500, 545, 592, 641, 692, 745, 800, 857, 916, 977, 1040, 1105, 1172, 1241, 1312, 1385, 1460, 1537, 1616, 1697, 1780, 1865, 1952, 2041, 2132, 2225, 2320, 2417, 2516

Offset: 0

Vincenzo Librandi, May 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A114962.

Cf. A000290, A243451 (primes).

a241751 = (+ 16) . (^ 2)  -- Reinhard Zumkeller, Apr 11 2015

Magma
```
[n^2+16: n in [0..60]];
```
Mathematica
```
Table[n^2 + 16, {n, 0, 60}]
```

a(n)=n^2+16 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (16 - 31*x + 17*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 2*n - 1.
From Amiram Eldar, Nov 03 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + 4*Pi*coth(4*Pi))/32.
Sum_{n>=0} (-1)^n/a(n) = (1 + 4*Pi*cosech(4*Pi))/32. (End)
E.g.f.: exp(x)*(16 + x + x^2). - Elmo R. Oliveira, Nov 29 2024

A243449 Primes of the form n^2 + 14.

Original entry on oeis.org

23, 239, 743, 1103, 2039, 5639, 7583, 8663, 27239, 33503, 38039, 42863, 59063, 81239, 88223, 91823, 119039, 131783, 140639, 164039, 189239, 205223, 245039, 263183, 288383, 328343, 342239, 378239, 393143, 400703, 431663, 439583, 514103, 660983, 710663, 950639

Offset: 1

Vincenzo Librandi, Jun 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A121250 (associated n).

Cf. primes of the form n^2+k: A144255 (k=1), A056899 (k=2), A049423 (k=3), A005473 (k=4), A056905 (k=5), A056909 (k=6), A079138 (k=7), A138338 (k=8), A138353 (k=9), A138355 (k=10), A138362 (k=11), A138368 (k=12), A138375 (k=13), this sequence (k=14), A243450 (k=15), A243451 (k=16), A228244 (k=17), A174812 (k=42).

[a: n in [0..1000] | IsPrime(a) where a is n^2+14];

Select[Table[n^2 + 14, {n, 0, 2000}], PrimeQ]
Select[Range[1,1001,2]^2+14,PrimeQ] (* Harvey P. Dale, May 30 2023 *)

A346145 Primes of the form k^2 + 25.

Original entry on oeis.org

29, 41, 61, 89, 281, 349, 509, 601, 701, 809, 1049, 1181, 1321, 1789, 2141, 2729, 3389, 4649, 5209, 5501, 5801, 8861, 9241, 9629, 10429, 11261, 11689, 12569, 15401, 15901, 17449, 17981, 18521, 19069, 21341, 21929, 23741, 24989, 26921, 27581, 33149, 39229, 40829, 41641, 42461, 45821, 46681, 52009

Offset: 1

Todor Szimeonov, Jul 06 2021

nonn

k^2 + 25 = (k+5i)*(k-5i), where i is the imaginary unit.

Todor Szimeonov, A dramatic encounter

Cf. A002144, A002496, A005473, A138353, A243451.

Select[Range[230]^2 + 25, PrimeQ] (* Amiram Eldar, Jul 06 2021 *)

list(lim)=my(v=List(),p); forstep(k=2,sqrtint(lim\1-25),2, if(isprime(p = k^2+25), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 06 2021

a(n) >> n log^2 n (Brun sieve). - Charles R Greathouse IV, Jul 06 2021

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A256839 Primes of form n^2 + 14641.

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A256840 Primes of form n^2 + 20736.

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A256841 Primes of form n^2 + 28561.

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A241751 a(n) = n^2 + 16.

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A243449 Primes of the form n^2 + 14.

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A346145 Primes of the form k^2 + 25.

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