A344123 -id:A344123

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A344124

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

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

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.

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)^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).

EXAMPLE

1.1545383304763889439221065945555168298987751974487...

CROSSREFS

Cf. A000040, A001481, A055025.

Cf. A344123, A344125.

KEYWORD

nonn,cons,more

AUTHOR

A.H.M. Smeets, May 09 2021

STATUS

approved

A344125

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

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn,cons,more

AUTHOR

A.H.M. Smeets, May 09 2021

STATUS

approved

A371012

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

LINKS

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.

FORMULA

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).

MATHEMATICA

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

PROG

(PARI) 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], ", "))); }

CROSSREFS

Cf. A001481, A006530, A344123.

KEYWORD

nonn,easy

AUTHOR

Amiram Eldar, Mar 08 2024

STATUS

approved

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