A344125 - OEIS

A344125

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

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

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OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..50.

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

FORMULA

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).

EXAMPLE

1.0685921056549901352029480207432436136133359081017...

CROSSREFS

Cf. A000040, A001481, A055025.

Cf. A344123, A344124.

Sequence in context: A296845 A030644 A319032 * A073462 A231095 A201195

Adjacent sequences: A344122 A344123 A344124 * A344126 A344127 A344128

KEYWORD

nonn,cons,more

AUTHOR

A.H.M. Smeets, May 09 2021

STATUS

approved