A243379 -id:A243379 - cp's OEIS frontend

A344123 Decimal expansion of Sum_{i > 0} 1/A001481(i)^2.

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

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an equivalent of zeta(2) (= Pi^2/6 = A013661), where Euler's zeta(2) is over all positive integers, with prime elements in A000040, while this constant is over all positive integers that can be written as the sum of two squares (A001481) with prime elements given in A055025.

Close to the value of e^(3/2)/Pi.

			1.4265560635125928786968093161550816361276693636770...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000040, A001481, A055025, A175647, A243379, A334448.

Cf. A344124, A344125.

Equals Sum_{i > 0} 1/A001481(i)^2.
Equals Product_{i > 0} 1/(1-A055025(i)^-2).
Equals 1/(1-prime(1)^(-2)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-2)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-4)), where prime(n) = A000040(n).
Equals (4/3)/(A243379*A334448).
Equals zeta_{2,0} (2) * zeta_{4,1} (2) * zeta_{4,3} (4), where zeta_{4,1} (2)  = A175647 and zeta_{2,0} (s) = 2^s/(2^s - 1).

A087691 Squares of primes of the form 4*k+3.

Original entry on oeis.org

9, 49, 121, 361, 529, 961, 1849, 2209, 3481, 4489, 5041, 6241, 6889, 10609, 11449, 16129, 17161, 19321, 22801, 26569, 27889, 32041, 36481, 39601, 44521, 49729, 51529, 57121, 63001, 69169, 73441, 80089, 94249, 96721, 109561, 120409, 128881

Offset: 1

Cino Hilliard, Sep 27 2003

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A002145, A243379, A243381.

Select[4*Range[0,100]+3,PrimeQ]^2 (* Harvey P. Dale, Sep 10 2012 *)

p4np3(n)= forprime(x=3,n,if(x%4==3,y=x*x; print1(y, ", ")));

a(n) = A002145(n)^2.
a(n) ~ 4n^2 * (log n)^2. - Charles R Greathouse IV, Sep 20 2016
From Amiram Eldar, Dec 02 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A243381.
Product_{n>=1} (1 - 1/a(n)) = A243379. (End)

More terms from Ray Chandler, Oct 26 2003

A066792 a(n) = phi(n^3 + n^2 + n + 1).

Original entry on oeis.org

1, 2, 8, 16, 64, 48, 216, 160, 288, 320, 1000, 480, 1344, 768, 1568, 1792, 4096, 1344, 4320, 2880, 4800, 3840, 8448, 3328, 11520, 7488, 12168, 6912, 17472, 6720, 24960, 13824, 16000, 13824, 25344, 14688, 46656, 19584, 26112, 24320, 64000, 19488

Offset: 0

Benoit Cloitre, Jan 18 2002

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000010 (phi), A053698, A243379.

Prepend[EulerPhi[Total[#^Range[0,3]]]&/@Range[45],1]  (* Harvey P. Dale, Feb 19 2011 *)

a(n) = eulerphi(n^3 + n^2 + n + 1); \\ Harry J. Smith, Mar 27 2010

a(n) = A000010(A053698(n)). - Michel Marcus, Sep 06 2022

Sum_{k=1..n} a(k) = c * n^4 + O((n*log(n))^3), where c = (3/16) * Product_{primes p == 1 (mod 4)} (1 - 3/p^2) * Product_{primes p == 3 (mod 4)} (1 - 1/p^2) = 0.13549316168... . - Amiram Eldar, Dec 09 2024

A340617 Decimal expansion of Product_{p prime, p == 3 (mod 4)} (1 - 2/p^2).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 13 2021

			0.7210979782407524158324311775035064193323800948822709044864277469512...

X. Gourdon and P. Sebah, Some Constants from Number theory
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, page 38 (case 4 3 2).

Cf. A002145, A065474, A085991, A243379, A335963.

Digits := 150;
with(NumberTheory);
DirichletBeta := proc(s) (Zeta(0, s, 1/4) - Zeta(0, s, 3/4))/4^s; end proc;
alfa := proc(s) DirichletBeta(s)*Zeta(s)/((1 + 1/2^s)*Zeta(2*s)); end proc;
beta := proc(s) (1 - 1/2^s)*Zeta(s)/DirichletBeta(s); end proc;
pzetamod43 := proc(s, terms) 1/2*Sum(Moebius(2*j + 1)*log(beta((2*j + 1)*s))/(2*j + 1), j = 0..terms); end proc;
evalf(exp(-Sum(2^t*pzetamod43(2*t, 70)/t, t = 1..200)));

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}];
Z2[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = 2^w * P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z2[4, 3, 2], digits]], 10, digits-1][[1]]

Equals 2*A065474/A335963.

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A344123 Decimal expansion of Sum_{i > 0} 1/A001481(i)^2.

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A087691 Squares of primes of the form 4*k+3.

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A066792 a(n) = phi(n^3 + n^2 + n + 1).

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A340617 Decimal expansion of Product_{p prime, p == 3 (mod 4)} (1 - 2/p^2).

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