A145824 -id:A145824 - cp's OEIS frontend

A080149 Numbers k such that k^2 + 1 and k^2 + 3 are both prime.

2, 4, 10, 14, 74, 94, 130, 134, 146, 160, 230, 256, 326, 340, 350, 406, 430, 440, 470, 584, 634, 686, 700, 704, 784, 860, 920, 986, 1054, 1070, 1156, 1210, 1324, 1340, 1354, 1366, 1394, 1420, 1456, 1460, 1564, 1700, 1784, 1816, 1876, 2006, 2080, 2096, 2174

Offset: 1

T. D. Noe, Jan 30 2003

Hardy and Littlewood conjecture that this sequence is infinite. This sequence is the intersection of A005574 (k such that k^2 + 1 is prime) and A049422 (k such that k^2 + 3 is prime).
From Jacques Tramu, Sep 10 2018: (Start)
a(10000)    =     2473624;  C = 2.91596513
a(100000)   =    35866246;  C = 2.70591741
a(1000000)  =   483764726;  C = 2.53454683
a(2000000)  =  1049178316;  C = 2.49209641
a(3000000)  =  1647417724;  C = 2.46880647
a(4000000)  =  2267125384;  C = 2.45259161
a(5000000)  =  2903162576;  C = 2.44036006
a(6000000)  =  3551848640;  C = 2.43024082
a(7000000)  =  4212006124;  C = 2.42214552
a(8000000)  =  4881390700;  C = 2.41510010
a(9000000)  =  5559542740;  C = 2.40915933
a(10000000) =  6245573750;  C = 2.40405768
a(20000000) = 13393786900;  C = 2.36959294
a(30000000) = 20908970800;  C = 2.35131696
a(40000000) = 28659267134;  C = 2.33835867
a(50000000) = 36590858294;  C = 2.32865934
C is the quotient a(n) / (n * log(n) * log(n)). (End)

			10 is in this sequence because 101 and 103 are both prime.

P. Ribenboim, "The New Book of Prime Number Records," Springer-Verlag, 1996, p. 408.

Zak Seidov, Table of n, a(n) for n = 1..32898 (terms < 10^7, first 1000 terms from T. D. Noe)
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.

Cf. A005574, A049422, A145824.

lst={}; Do[If[PrimeQ[m^2+1]&&PrimeQ[m^2+3], AppendTo[lst, m]], {m, 3000}]; lst
okQ[n_]:=Module[{n2=n^2},PrimeQ[n2+1]&&PrimeQ[n2+3]]; Select[Range[2200], okQ]  (* Harvey P. Dale, Apr 21 2011 *)
Select[Range[2500],AllTrue[#^2+{1,3},PrimeQ]&] (* Harvey P. Dale, Sep 07 2023 *)

isA080149(n) = isprime(n^2+1) && isprime(n^2+3) \\ Michael B. Porter, Mar 22 2010

Conjecture: a(n) is asymptotic to c*n*log(n)^2 with c around 2.9... - Benoit Cloitre, Apr 16 2004

A200992 Primes p of form n^2 + 1 such that p+2 and p+6 are also prime.

Original entry on oeis.org

5, 17, 101, 5477, 21317, 65537, 193601, 220901, 341057, 846401, 1144901, 1336337, 1752977, 1943237, 2016401, 4326401, 6100901, 6760001, 11062277, 14032517, 23001617, 35952017, 41731601, 51265601, 55741157, 79103237, 83978897, 91278917, 92083217, 110040101

Offset: 1

Zak Seidov, Nov 25 2011

nonn

Subsequence of A145824 which itself is subsequence of A002496.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A002496, A145824.

Select[Range[11000]^2+1,AllTrue[#+{0,2,6},PrimeQ]&] (* Harvey P. Dale, Jul 21 2024 *)

A358342 Lesser of twin primes p such that sigma((p-1)/2) + tau((p-1)/2) is a prime.

Original entry on oeis.org

3, 5, 17, 65537, 1927561217, 6015902625062501, 12370388895062501, 835920078368222501, 6448645485213008897, 50973659693056000001, 54332889713542767617, 64304984013657011717, 112112769248058062501, 147337258721536000001

Offset: 1

Jaroslav Krizek, Nov 10 2022

Lesser of twin primes p such that A000203((p-1)/2) + A000005((p-1)/2) is a prime q.
The first 4 terms are Fermat primes from A019434.
Corresponding values of primes q:  2, 5, 19, 65551, 2248681529, ...
Subsequence of A272060 and A272061.
Lesser of twin primes of the form 2*m+1 with m a term of A064205.
There are no other terms <= 10^14.
All the terms above 3 are in A145824. - Amiram Eldar, Jan 05 2023

			17 and 19 are twin primes; sigma((17-1)/2) + tau((17-1)/2) = sigma(8) + tau(8) = 15 + 4 = 19; 19 is prime, so 17 is in the sequence.

Intersection of A001359 and A272061.

Cf. A000005 (tau), A000203 (sigma), A019434, A064205, A145824, A272060.

[n: n in [3..10^7] | IsPrime(n) and IsPrime(n+2) and IsPrime(&+Divisors((n-1) div 2) + #Divisors((n-1) div 2))]

Join[{3}, Select[4*Range[25000]^2 + 1, PrimeQ[#] && PrimeQ[# + 2] && PrimeQ[DivisorSigma[1, (# - 1)/2] + DivisorSigma[0, (# - 1)/2]] &]]
(* or *)
A272061 = Cases[Import["https://oeis.org/A272061/b272061.txt", "Table"], {, }][[;; , 2]]; Select[A272061, PrimeQ[# + 2] &] (* Amiram Eldar, Jan 05 2023 *)

isok(p) = if (isprime(p) && isprime(p+2), my(f=factor((p-1)/2)); isprime(sigma(f)+numdiv(f))); \\ Michel Marcus, Nov 23 2022

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A080149 Numbers k such that k^2 + 1 and k^2 + 3 are both prime.

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A200992 Primes p of form n^2 + 1 such that p+2 and p+6 are also prime.

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A358342 Lesser of twin primes p such that sigma((p-1)/2) + tau((p-1)/2) is a prime.

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