A108814 -id:A108814 - cp's OEIS frontend

A096012 Numbers k such that k^2+1 and (k+2)^2+1 are both prime; twin k^2+1 primes.

2, 4, 14, 24, 54, 124, 204, 384, 464, 634, 644, 714, 1094, 1144, 1174, 1244, 1274, 1314, 1374, 1564, 1614, 1674, 1684, 1964, 2054, 2084, 2094, 2404, 2454, 2534, 2664, 2834, 2924, 3134, 3304, 3534, 3754, 3774, 4024, 4154, 4174, 4364, 4604, 4614, 4734, 4784

Offset: 1

Jason Earls, Jul 20 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Cf. A005574, A108814.

[n: n in [1..5000] | IsPrime(n^2+1) and IsPrime((n+2)^2+1)]; // Vincenzo Librandi, Feb 27 2016

Select[Range[5000],AllTrue[{#^2+1,(#+2)^2+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 23 2014 *)
Select[Range[5000], PrimeQ[#^2 + 1] && PrimeQ[(# + 2)^2 + 1] &] (* Vincenzo Librandi, Feb 27 2016 *)

isok(n) = isprime(n^2+1) && isprime((n+2)^2+1); \\ Michel Marcus, Feb 27 2016

a(k) = A108814(k) - 1. - Jeppe Stig Nielsen, Feb 26 2016

A108868 Numbers n such that n^5 + 3 is semiprime.

Original entry on oeis.org

1, 2, 4, 6, 11, 14, 18, 19, 24, 31, 32, 38, 40, 46, 50, 55, 59, 70, 74, 76, 84, 92, 96, 100, 115, 119, 128, 139, 148, 150, 151, 154, 155, 158, 164, 178, 184, 200, 203, 204, 206, 210, 230, 236, 238, 239, 242, 248, 256, 263, 272, 278, 284, 295, 299, 304, 306, 310

Offset: 1

Jonathan Vos Post, Jul 12 2005

Note that n^5 + 3 is irreducible over integers, unlike n^5 + 1 as in A104238.

			1^5 + 3 = 4 = 2 * 2
2^5 + 3 = 35 = 5 * 7
4^5 + 3 = 1027 = 13 * 79
6^5 + 3 = 7779 = 3 * 2593
11^5 + 3 = 161054 = 2 * 80527
14^5 + 3 = 89 * 6043
100^5 + 3 = 10000000003 = 7 * 1428571429
1000^5 + 3 = 1000000000000003 = 14902357 * 67103479
1000000^5 + 3 = 1000000000000000000000000000003 = 1859827 * 537684419034673655130289.

Cf. A104238, A108814.

with(numtheory): a:=proc(n) if bigomega(n^5+3)=2 then n else fi end: seq(a(n),n=1..400); # Emeric Deutsch, Jul 16 2005

Select[Range[400],PrimeOmega[#^5+3]==2&] (* Harvey P. Dale, Jul 16 2017 *)

More terms from Emeric Deutsch, Jul 16 2005

A217583 Numbers n^2+1 such that (n-1)^2+1 and (n+1)^2+1 are prime.

Original entry on oeis.org

10, 26, 226, 626, 3026, 15626, 42026, 148226, 216226, 403226, 416026, 511226, 1199026, 1311026, 1380626, 1550026, 1625626, 1729226, 1890626, 2449226, 2608226, 2805626, 2839226, 3861226, 4223026, 4347226, 4389026, 5784026, 6027026, 6426226, 7102226, 8037226

Offset: 1

Michel Lagneau, Oct 07 2012

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002496, A108814.

with(numtheory):for n from 1 to 500 do: x1:=n^2+1:x2:=(n+2)^2 + 1:if type(x1,prime)=true and type(x2,prime)=true then printf(`%d, `,(n+1)^2+1):else fi:od:

Select[Partition[Range[3000]^2+1,3,1],AllTrue[{#[[1]],#[[3]]},PrimeQ]&][[All,2]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 03 2019 *)

forstep(n=3,1e3,2,if(isprime(n^2+2*n+2) && isprime(n^2-2*n+2), print1(n^2+1", "))) \\ Charles R Greathouse IV, Oct 09 2012

A351170 Consider the primes of the form p(m)=m^2+1 such that p(m+2) is also prime for some m. The sequence lists the sums p(m) + p(m+2).

Original entry on oeis.org

22, 54, 454, 1254, 6054, 31254, 84054, 296454, 432454, 806454, 832054, 1022454, 2398054, 2622054, 2761254, 3100054, 3251254, 3458454, 3781254, 4898454, 5216454, 5611254, 5678454, 7722454, 8446054, 8694454, 8778054, 11568054, 12054054, 12852454, 14204454, 16074454

Offset: 1

Michel Lagneau, Feb 04 2022

nonn

			a(3) = 454 because A096012(3) = 14, 14^2+1 = 197, (14+2)^2+1 = 257, and 197 + 257 = 454.

Michel Lagneau, a(n)==54 (mod 100)

Cf. A002496, A005574, A096012, A108814, A206328.

nn:=3000:
for n from 2 by 2 to nn do:
  p1:=n^2+1:p2:=(n+2)^2+1:
   if isprime(p1) and isprime(p2)
    then
    s:=p1+p2:printf(`%d, `,s):
    else
   fi:
od:

f[n_] := 2*n^2 + 4*n + 6; f /@ Select[Range[3000], And @@ PrimeQ[{#^2 + 1, (# + 2)^2 + 1}] &] (* Amiram Eldar, Feb 04 2022 *)

lista(nn) = {for (m=1, nn, if (isprime(m^2+1) && isprime(m^2+4*m+5), print1(2*m^2+4*m+6, ", ")););} \\ Michel Marcus, Feb 04 2022

For n>1, a(n) == 54 (mod 100) (see proof above).
a(n) = 2*(A096012(n)+1)^2+4 = 2*A108814(n)^2+4. - Alois P. Heinz, Feb 04 2022
For n > 1, a(n) mod 400 = 54; a(n) mod 1200 = 54 or 454; a(n) mod 2000 = 54, 454, or 1254; a(n) mod 54, 454, 1254, or 2454. - Jon E. Schoenfield, Feb 04 2022

cp's OEIS Frontend

A096012 Numbers k such that k^2+1 and (k+2)^2+1 are both prime; twin k^2+1 primes.

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A108868 Numbers n such that n^5 + 3 is semiprime.

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A217583 Numbers n^2+1 such that (n-1)^2+1 and (n+1)^2+1 are prime.

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A351170 Consider the primes of the form p(m)=m^2+1 such that p(m+2) is also prime for some m. The sequence lists the sums p(m) + p(m+2).

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