A344124 -id:A344124 - 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).

A344125 Decimal expansion of Sum_{i > 0} 1/A001481(i)^4.

Original entry on oeis.org

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

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an analog of zeta(4) (= Pi^4/90 = A013662), where Euler's zeta(4) 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.

			1.0685921056549901352029480207432436136133359081017...

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.

Cf. A344123, A344124.

Equals Sum_{i > 0} 1/A001481(i)^4.
Equals Product_{i > 0} 1/(1-A055025(i)^-4).
Equals 1/(1-prime(1)^(-4)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-4)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-8)), where prime(n) = A000040(n).
Equals zeta_{2,0} (4) * zeta_{4,1} (4) * zeta_{4,3} (8), where zeta_{2,0} (s) = 2^s/(2^s - 1).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula