A344123 -id:A344123 - cp's OEIS frontend

A344124 Decimal expansion of Sum_{i > 0} 1/A001481(i)^3.

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

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an analog of zeta(3) (= Apéry's constant = A002117), where Euler's zeta(3) 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.

			1.1545383304763889439221065945555168298987751974487...

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, A344125.

Equals Sum_{i > 0} 1/A001481(i)^3.
Equals Product_{i > 0} 1/(1-A055025(i)^-3).
Equals 1/(1-prime(1)^(-3)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-3)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-6)), where prime(n) = A000040(n).
Equals zeta_{2,0} (3) * zeta_{4,1} (3) * zeta_{4,3} (6), where 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).

A371012 The largest prime that divides the n-th number that is the sum of 2 squares; a(2) = 1.

Original entry on oeis.org

1, 2, 2, 5, 2, 3, 5, 13, 2, 17, 3, 5, 5, 13, 29, 2, 17, 3, 37, 5, 41, 5, 7, 5, 13, 53, 29, 61, 2, 13, 17, 3, 73, 37, 5, 3, 41, 17, 89, 5, 97, 7, 5, 101, 13, 53, 109, 113, 29, 13, 11, 61, 5, 2, 13, 17, 137, 3, 29, 73, 37, 149, 17, 157, 5, 3, 41, 13, 17, 173, 89

Offset: 2

Amiram Eldar, Mar 08 2024

Amiram Eldar, Table of n, a(n) for n = 2..10001
Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.

Cf. A001481, A006530, A344123.

FactorInteger[#][[-1, 1]] & /@ Select[Range[200], SquaresR[2, #] > 0 &]

lista(kmax) = {my(f, is); print1(1, ", "); for(k = 2, kmax, f = factor(k); is = 1; for(i=1, #f~, if(f[i, 2]%2 && f[i, 1]%4 == 3, is = 0; break)); if(is, print1(f[#f~, 1], ", ")));}

a(n) = A006530(A001481(n+1)).

Sum_{k >= 2, A001481(k) < n} a(k) = (1/4) * c * n^2/log(n) + o(n^2/log(n)), where c = A344123 (Jakimczuk, 2024, Theorem 4.9, p. 54).

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A344124 Decimal expansion of Sum_{i > 0} 1/A001481(i)^3.

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

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A371012 The largest prime that divides the n-th number that is the sum of 2 squares; a(2) = 1.

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