A131653 -id:A131653 - cp's OEIS frontend

A117543 Decimal expansion of the sum of the reciprocals of squared semiprimes.

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

Offset: 0

T. D. Noe, Mar 28 2006

Geoffrey Landis and Jonathan Vos Post (personal communication) show that this constant equals ((P2)^2 + P4)/2, where P2 and P4 are constants in A085548 and A085964, respectively.

			0.14076043434902338822275... = 1/4^2 + 1/6^2 + 1/9^2 + 1/10^2 + 1/14^2 +1/15^2 +...

Richard J. Mathar, Series of Reciprocal Powers of k-almost Primes, arXiv:0803.0900 [math.NT], 2008-2009. See Table 4.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A001358 (semiprimes), A154928 (derivative).

Cf. A131653 (squared triprimes). [T. D. Noe, Oct 09 2008]

RealDigits[(PrimeZetaP[2]^2 + PrimeZetaP[4])/2, 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)

A282468 Decimal expansion of the zeta function at 2 of every second prime number.

Original entry on oeis.org

1, 4, 4, 7, 1, 5, 5, 8, 6, 6, 8, 8, 7

Offset: 0

Terry D. Grant, Apr 14 2017

From Husnain Raza, Aug 30 2023: (Start)
Note that since p_n > n*log(n), we can place a bound on the tail of the sum:
Sum_{n >= N} (prime(2n))^(-2) <= Sum_{n >= N} (2*n*log(2n))^(-2) <= Integral_{x=N..oo} (2*x*log(2x))^(-2) dx.
Taking the sum over all primes < 10^12, we see that the constant lies between 0.14471558668870 and 0.14471558668873. (End)

			1/3^2 + 1/7^2 + 1/13^2 + 1/19^2 + 1/29^2 + ... = 0.14471558...

Husnain Raza, PARI program for calculating more digits.

Cf. A031215, A085548.

Zeta functions at 2: A085548 (for primes), A275647 (for nonprimes), A013661 (for natural numbers), A117543 (for semiprimes), A131653 (for triprimes), A222171 (for even numbers), A111003 (for odd numbers).

PARI
```
sum(n=1, 2500000, 1./prime(2*n)^2)
```
PARI
```
\\ see Raza link
```

Equals Sum_{n>=1} 1/A031215(n)^2 = Sum_{n>=1} 1/prime(2n)^2.

a(8)-a(12) from Husnain Raza, Aug 31 2023

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A117543 Decimal expansion of the sum of the reciprocals of squared semiprimes.

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