A096012 -id:A096012 - cp's OEIS frontend

A337606 Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.

4, 8, 7, 6, 2, 2, 7, 7, 8, 1, 1, 1, 5, 7, 1, 7, 6, 8, 6, 1, 1, 6, 4, 6, 3, 9, 1, 4, 5, 2, 3, 8, 8, 4, 2, 3, 1, 3, 1, 6, 7, 7, 1, 2, 4, 4, 2, 9, 7, 3, 5, 7, 6, 3, 7, 7, 0, 1, 8, 1, 5, 8, 2, 9, 7, 2, 3, 6, 5, 6, 9, 0, 3, 4, 5, 4, 0, 0, 9, 2, 3, 4, 9, 8, 1, 0, 6, 6, 6, 1, 7, 4, 6, 4, 8, 5, 1, 9, 1, 4, 3, 3, 2, 8, 4, 1

Offset: 0

Amiram Eldar, Sep 04 2020

The name of this constant was suggested by Finch (2003).
Gaussian twin primes on the line x + i in the complex plane are Gaussian primes pair of the form (m - 1 + i, m + 1 + i). The numbers m are numbers such that (m-1)^2 + 1 and (m+1)^2 + 1 are both primes (A096012 plus 1).
Shanks (1960) conjectured that the number of these pairs with m <= x is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section and evaluated it by 0.4876.
The first 100 digits of 4*c were calculated by Ettahri et al. (2019).

			0.487622778111571768611646391452388423131677124429735...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 90.

Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Section 8).
Daniel Shanks, , A note on Gaussian twin primes, Mathematics of Computation, Vol. 14, No. 70 (1960), pp. 201-203.

Cf. A002496, A005574, A096012, A206328.

Similar constants: A005597, A331941, A337607, A337608.

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi^2/8 * Zs[4, 1, 4]/Z[4, 1, 2]^2, digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Equals (Pi^2/8) * Product_{primes p == 1 (mod 4)} (1 - 4/p)*((p + 1)/(p - 1))^2.

More digits from Vaclav Kotesovec, Jan 15 2021

A206328 Primes of the form n^2+1 such that (n+2)^2+1 is also prime.

Original entry on oeis.org

5, 17, 197, 577, 2917, 15377, 41617, 147457, 215297, 401957, 414737, 509797, 1196837, 1308737, 1378277, 1547537, 1623077, 1726597, 1887877, 2446097, 2604997, 2802277, 2835857, 3857297, 4218917, 4343057, 4384837, 5779217, 6022117, 6421157, 7096897, 8031557

Offset: 1

Michel Lagneau, Feb 06 2012

Primes corresponding to A096012 and subset of A002496.
For n > 1, a(n) ==7 (mod 10) because n ==4 (mod 10).
Conjecture: this sequence is infinite.

			For n = 4, n^2 + 1 = 17 is prime and  (n+2)^2 + 1 = 37 is also prime => 17 is in the sequence.

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

Cf. A096012, A002496.

for n from 1 to 4000 do: x:=n^2+1:y:=(n+2)^2+1:if type(x,prime)=true and type(y,prime)=true then printf(`%d, `,x): else fi:od:

Select[Partition[Range[3000]^2+1,3,1],AllTrue[{#[[1]],#[[3]]},PrimeQ]&][[All,1]] (* Harvey P. Dale, Jan 16 2023 *)

A108814 Numbers k such that k^4 + 4 is semiprime.

Original entry on oeis.org

3, 5, 15, 25, 55, 125, 205, 385, 465, 635, 645, 715, 1095, 1145, 1175, 1245, 1275, 1315, 1375, 1565, 1615, 1675, 1685, 1965, 2055, 2085, 2095, 2405, 2455, 2535, 2665, 2835, 2925, 3135, 3305, 3535, 3755, 3775, 4025, 4155, 4175, 4365, 4605, 4615, 4735, 4785

Offset: 1

Jason Earls, Jul 10 2005

nonn

Except for the first, all the terms above generate brilliant numbers.

Numbers n such that n - 1 + i and n + 1 + i are (twin) Gaussian primes, see Shanks. - Charles R Greathouse IV, Apr 20 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000
Daniel Shanks, A Note on Gaussian Twin Primes, Mathematics of Computation 14:70 (1960), pp. 201-203.

Cf. A001358, A057781, A096012.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [ n: n in [1..5000] | IsSemiprime(n^4+4)]; // Vincenzo Librandi, Apr 20 2011

Select[Range[5000],PrimeOmega[#^4+4]==2&] (* Harvey P. Dale, Sep 07 2017 *)

forstep(n=1,1e5,2,if(isprime(n^2-2*n+2) && isprime(n^2+2*n+2), print1(n", "))) \\ Charles R Greathouse IV, Apr 20 2011

a(k) = A096012(k) + 1. (Because n^4+4 = ((n-1)^2+1)((n+1)^2+1).) - Jeppe Stig Nielsen, Feb 26 2016

A302021 Numbers k such that k^2+1, (k+2)^2+1 and (k+6)^2+1 are prime.

Original entry on oeis.org

4, 14, 124, 204, 464, 1144, 1314, 1564, 1964, 2454, 3134, 4174, 4364, 5584, 5874, 6234, 7804, 8174, 8784, 9874, 9894, 10424, 12354, 12484, 12874, 14034, 14194, 15674, 16224, 18274, 18994, 21134, 21344, 22344, 22624, 23134, 23784, 23944, 24974, 25554, 26504, 26934, 27064, 27804, 29364

Offset: 1

Seiichi Manyama, Mar 31 2018

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..200 from Seiichi Manyama)

Cf. A005574, A096012, A302087.

[n: n in [1..30000] | IsPrime(n^2+1) and IsPrime((n+2)^2+1) and IsPrime((n+6)^2+1)]; // Vincenzo Librandi, Apr 02 2018

select(k->isprime(k^2+1) and isprime((k+2)^2+1) and isprime((k+6)^2+1),[$1..40000]); # Muniru A Asiru, Apr 02 2018

Select[Range[1, 30000], PrimeQ[#^2 + 1] && PrimeQ[(# + 2)^2 + 1] && PrimeQ[(# + 6)^2 + 1] &] (* Vincenzo Librandi, Apr 02 2018 *)

isok(k) = isprime(k^2+1) && isprime((k+2)^2+1) && isprime((k+6)^2+1); \\ Altug Alkan, Apr 02 2018

from sympy import isprime
k, klist, A302021_list = 0, [isprime(i**2+1) for i in range(6)], []
while len(A302021_list) < 10000:
    i = isprime((k+6)**2+1)
    if klist[0] and klist[2] and i:
        A302021_list.append(k)
    k += 1
    klist = klist[1:] + [i] # Chai Wah Wu, Apr 01 2018

A302087 Numbers k such that k^2+1 and (k+6)^2+1 are both prime.

Original entry on oeis.org

4, 10, 14, 20, 84, 110, 120, 124, 150, 170, 204, 224, 230, 250, 264, 300, 400, 430, 464, 490, 570, 674, 680, 690, 930, 960, 1004, 1054, 1060, 1140, 1144, 1150, 1314, 1410, 1434, 1550, 1564, 1570, 1580, 1654, 1784, 1870, 1964, 1974, 2050, 2074, 2080, 2120, 2260, 2304, 2314

Offset: 1

Seiichi Manyama, Mar 31 2018

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A005574, A023201, A096012, A302021.

[n: n in [1..2500] | IsPrime(n^2+1) and IsPrime((n+6)^2+1)]; // Vincenzo Librandi, Apr 02 2018

select(k->isprime(k^2+1) and isprime((k+6)^2+1),[$1..3000]); # Muniru A Asiru, Apr 02 2018

Select[Range[3000], PrimeQ[#^2 + 1] && PrimeQ[(# + 6)^2 + 1]&] (* Vincenzo Librandi, Apr 02 2018 *)

isok(k) = isprime(k^2+1) && isprime((k+6)^2+1); \\ Altug Alkan, Apr 02 2018

from sympy import isprime
k, klist, A302087_list = 0, [isprime(i**2+1) for i in range(6)], []
while len(A302087_list) < 10000:
    i = isprime((k+6)**2+1)
    if klist[0] and i:
        A302087_list.append(k)
    k += 1
    klist = klist[1:] + [i] # Chai Wah Wu, Apr 01 2018

A339007 Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are prime numbers with q - p square.

Original entry on oeis.org

24, 6, 312984, 16896, 120, 734994, 10640, 10, 1946016, 150, 171864, 180, 31200, 17136, 120, 84, 8976, 54, 137256, 300, 231504, 66, 184, 360126, 24, 5824, 2496, 224, 261696, 90, 4359344, 66, 50160, 68816, 280, 864, 1524696, 570, 219336, 11520, 8487984, 126, 22704

Offset: 1

Michel Lagneau, Nov 18 2020

nonn

4*n*(k + n) is a square. If n is a square, then k + n is also a square.
If n is prime, then n divides k.
If we add the additional condition that p and q are two consecutive primes of the form m^2 + 1, then we obtain the sequence A339008, with A339008(n) = a(n) for n = 1, 2, 3, 4, 6, 7 and 9.

			a(1) = 24 because 24^2 + 1 = 577, (24 + 2)^2 + 1 = 677 and 677 - 577 = 10^2 is a square. The other values m such that p = m^2 + 1 and q = (m+2)^2 + 1 are primes with q - p square are 11024, 133224, 156024, 342224, 416024,...
a(2) = 6 because 6^2 + 1 = 37, (6 + 4)^2 + 1 = 101 and 101 - 37 = 8^2 is a square. The other values m such that p = m^2 + 1 and q = (m+4)^2 + 1 are primes with q - p square are 16, 126, 1350, 1456, 1566, 2310, 5200,...

Cf. A002496, A096012, A193558, A206328, A216330, A339008.

for n from 1 to 50 do:
ii:=0:
for k from 2 by 2 to 10^9 while(ii=0) do:
   p:=k^2+1:q:=(k+2*n)^2 +1:
    if isprime(p) and isprime(q) and sqrt(q-p)=floor(sqrt(q-p))
     then
     ii:=1:printf(`%d %d \n`,n,k):
     else
    fi:
  od:
od:

a(n) = my(k=1); while (!(isprime(p=k^2+1) && isprime(q=(k+2*n)^2 + 1) && issquare(q-p)), k++); k; \\ Michel Marcus, Nov 18 2020

A339008 Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are two consecutive prime numbers of the same form with q - p square.

Original entry on oeis.org

24, 6, 312984, 16896, 240, 734994, 10640, 10360, 1946016, 2550, 13189264, 72996, 416520, 2184336, 1584360, 202484, 232696, 1700150, 2394456, 375360, 8736504, 9237866, 53629744, 360126, 87000, 574339974, 82404216, 23237760, 1249877496, 826650, 127119344, 1527720

Offset: 1

Michel Lagneau, Nov 18 2020

nonn

4*n*(k + n) is a square. If n is a square, then k + n is also a square.
If n is prime, then n divides k.
a(n) = A339007(n) for n = 1, 2, 3, 4, 6, 7 and 9.

			a(1) = 24 because 24^2 + 1 = 577, (24 + 2)^2 + 1 = 677. The numbers 577 and 677 are two consecutive primes of the form m^2+1, and 677 - 577 = 10^2 is a square. The other values m such that p = m^2 + 1 and q = (m+2)^2 + 1 are consecutive primes with q - p square are 11024, 133224, 156024, 342224, 416024, ...
a(2) = 6 because 6^2 + 1 = 37, (6 + 4)^2 + 1 = 101. The numbers 37 and 101 are two consecutive primes of the form m^2+1, and 101 - 37 = 8^2 is a square. The other values m such that p = m^2 + 1 and q = (m+4)^2 + 1 are consecutive primes with q - p square are 16, 126, 1350, 1456, 1566, 2310, 5200, ...

Chai Wah Wu, Table of n, a(n) for n = 1..60

Cf. A002496, A096012, A193558, A206328, A216330, A339007.

for n from 1 to 25 do:
ii:=0:n1:=0:q:=2:
  for k from 2 by 2 to 10^9 while(ii=0) do:
    p:=k^2+1:
   if isprime(p)
    then
     x:=p-q:q:=p:z:=sqrt(x):
      if z=floor(z) and k-n1=2*n
       then
        ii:=1:printf(`%d %d \n`,n,n1):
         else
         n1:=k:
       fi:
    fi:
  od:
od:

consecutive(p, q) = {forprime(r = nextprime(p+1), precprime(q-1), if (isprime(r) && issquare(r-1), return(0));); return(1);}
a(n) = my(k=1); while (!(isprime(p=k^2+1) && isprime(q=(k+2*n)^2 + 1) && issquare(q-p) && consecutive(p, q)), k++); k; \\ Michel Marcus, Nov 30 2020

a(26)-a(32) from Chai Wah Wu, Dec 06 2020

A268867 Number of integers of the form m^2+1 between two consecutive pairs of primes of the same form.

Original entry on oeis.org

0, 7, 7, 27, 67, 77, 177, 77, 167, 7, 67, 377, 47, 27, 67, 27, 37, 57, 187, 47, 57, 7, 277, 87, 27, 7, 307, 47, 77, 127, 167, 87, 207, 167, 227, 217, 17, 247, 127, 17, 187, 237, 7, 117, 47, 7, 157, 57, 37, 197, 217, 87, 17, 137, 147, 287, 67, 547, 37, 187, 787

Offset: 1

Michel Lagneau, Feb 15 2016

nonn

Or number of integers of the form m^2+1 between two consecutive twin k^2+1 primes.
a(n)==7 mod 10 for n>1.
The primes of the sequence are 7, 17, 37, … (A030432), subsequence of A017353.
Conjecture 1: the sequence is infinite.
Conjecture 2: for n>1, the sequence sorted in ascending order with distinct values generates the set B = {b(k)} = {7, 17, 27, 37, …} = {7 + 10k}, k = 0,1,2,…  and {a(n)}/qZ = B/qZ with q = 2^m, m = 1, 2,… is a multiplicative group.
This has been verified for n up to 10^8.
A remarkable simple way to perceive properties of invariability in this sequence (or other particular sequences) is the use of finite groups of integers modulo q. This study can provide interesting interpretations for some regularities which describe properties in other mathematical spaces.
However, this concept is based on a conjecture if it is impossible to prove the infinity character of a sequence. This requires, for the calculations, a large number of elements in the sequence.
The groups of integers modulo 2^m are:
q = 2 => B/2Z = {1}, the trivial group.
q = 4 => B/4Z = {1,3}, the cyclic group of order 2 with two elements.
q = 8 => B/8Z = {1,3,5,7}, the group of order 4 with generating set {3,7} => the Klein four-group of order 2. The square of each element of B/8Z is 1. The group is not cyclic.
q = 16 => B/16Z = {1,3,5,7,9,11,13,15}, the group of order 8 with generating set {3,15}. The powers of 3 {1,3,9,11} are a subgroup of order 4, as are the powers of 5, {1,5,9,13}. The group B/16Z is not cyclic.
For higher powers q = 2^k, k>2,  B/(2^k)Z = {1,3,5,…,2^k-1}, with generating set {3, 2^k-1}. The group B/(2^k)Z is not cyclic.
The order of the group is given by Euler’s totient function (A000010): this is the product of the orders of the cyclic groups in the direct product (see the links).

			a(1)=0 because there exists 0 number of the form m^2+1 between the two consecutive pairs of primes(2^2+1, 4^2+1) and (4^2+1, 6^2+1);
a(2) = 7 because there exists 7 numbers of the form m^2+1 between the two consecutive pairs of primes(4^2+1, 6^2+1) and (14^2+1, 16^2+1): 50, 65, 82, 101, 122, 145 and 170.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Finite Group
Wikipedia, Finite group
Wikipedia, Multiplicative group of integers modulo n

Cf. A005574, A030432, A017353, A096012, A206328.

nn:=10000:T:=array(1..200):kk:=0:
for n from 4 by 2 to nn do:
   p1:=n^2+1:p2:=(n+2)^2+1:
    if isprime(p1) and isprime(p2)
     then
     kk:=kk+1:T[kk]:=n:
     else
    fi:
  od:
    for m from 1 to kk-1 do:
      q:=T[m+1]-T[m]-3:printf(`%d, `,q):
    od:

lst={};Do[If[PrimeQ[n^2+1],AppendTo[lst,n]],{n,1,10000}];Module[{tr=Transpose[Select[Partition[lst,2,1],#[[2]]-#[[1]]==2&]],fir,las},fir=Rest[tr[[1]]];las=Most[tr[[2]]];Flatten[Abs[Differences/@Thread[{fir,las}]]]-1/.{-1->0}]

A293621 Numbers k such that (2k)^2 + 1 and (2k+2)^2 + 1 are both primes.

Original entry on oeis.org

1, 2, 7, 12, 27, 62, 102, 192, 232, 317, 322, 357, 547, 572, 587, 622, 637, 657, 687, 782, 807, 837, 842, 982, 1027, 1042, 1047, 1202, 1227, 1267, 1332, 1417, 1462, 1567, 1652, 1767, 1877, 1887, 2012, 2077, 2087, 2182, 2302, 2307, 2367, 2392, 2397, 2477, 2507

Offset: 1

Amiram Eldar, Oct 13 2017

nonn

Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite. He gives the 24 terms below 10^3.

Sierpiński noted that the only triple of consecutive primes of the form (2n)^2 + 1 are for n = 1 (i.e., 1 and 2 are the only consecutive terms in this sequence), since every triple of consecutive terms contains at least one term which is divisible by 5.

			1 is in the sequence since (2*1)^2 + 1 = 5 and (2*1+2)^2 + 1 = 17 are both primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wacław Sierpiński, Remarque sur la distribution de nombres premiers, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 77-78.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Wikipedia, Schinzel's hypothesis H.

Subsequence of A001912.

Cf. A002496, A002522, A005574, A053755, A096012.

Select[Range[10^4], AllTrue[{(2#)^2+1, (2#+2)^2+1}, PrimeQ] &]

isok(n) = isprime((2*n)^2 + 1) && isprime((2*n+2)^2 + 1); \\ Michel Marcus, Oct 13 2017

a(n) = A096012(n)/2. - Amiram Eldar, Feb 24 2020

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

A337606 Decimal expansion of the Gaussian twin prime constant: the Hardy-Littlewood constant for A096012.

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A206328 Primes of the form n^2+1 such that (n+2)^2+1 is also prime.

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A108814 Numbers k such that k^4 + 4 is semiprime.

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A302021 Numbers k such that k^2+1, (k+2)^2+1 and (k+6)^2+1 are prime.

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A302087 Numbers k such that k^2+1 and (k+6)^2+1 are both prime.

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A339007 Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are prime numbers with q - p square.

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A339008 Least k such that p = k^2 + 1 and q = (k+2n)^2 + 1 are two consecutive prime numbers of the same form with q - p square.

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A293621 Numbers k such that (2k)^2 + 1 and (2k+2)^2 + 1 are both primes.