A275647 -id:A275647 - cp's OEIS frontend

A062312 Nonprime numbers squared.

1, 16, 36, 64, 81, 100, 144, 196, 225, 256, 324, 400, 441, 484, 576, 625, 676, 729, 784, 900, 1024, 1089, 1156, 1225, 1296, 1444, 1521, 1600, 1764, 1936, 2025, 2116, 2304, 2401, 2500, 2601, 2704, 2916, 3025, 3136, 3249, 3364, 3600, 3844, 3969, 4096, 4225

Offset: 1

Jason Earls, Jul 05 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A002808, A018252, A275647.

Select[Range[100],!PrimeQ[#]&]^2 (* Harvey P. Dale, Jan 27 2016 *)

for(n=1,30, if(isprime(n), n+1,print(n^2)))

n=0; for (m=1, 2000, if (!isprime(m), write("b062312.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 04 2009

is(n)=issquare(n,&n) && !isprime(n) \\ Charles R Greathouse IV, Sep 18 2015

a(n) = A018252(n)^2. - Omar E. Pol, Oct 30 2007

Sum_{n>=1} 1/a(n) = A275647. - Amiram Eldar, Oct 14 2020

More terms from Larry Reeves (larryr(AT)acm.org), Jul 05 2001

A278419 Decimal expansion of sum of cubes of reciprocals of nonprime numbers.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Nov 21 2016

			1.0272942638601507489766248468457432897895741704143495919035995386402...

Eric Weisstein's World of Mathematics, Zeta Function.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A275647.

RealDigits[Zeta[3] - PrimeZetaP[3], 10, 104][[1]]

zeta(3) - sumeulerrat(1/p, 3) \\ Amiram Eldar, Mar 19 2021

Sum_{n>=1} 1/n^3 - Sum_{n>=1} 1/prime(n)^3.
Equals zeta(3) - primezetaP(3).
Sum of cubes of reciprocals of composite numbers = zeta(3) - primezetaP(3) - 1 = 0.02729426386...

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

A335589 Decimal expansion of the sum of the reciprocals of the squares of composite numbers.

Original entry on oeis.org

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

Offset: 0

Jon E. Schoenfield, Jan 26 2021

			Equals 1/4^2 + 1/6^2 + 1/8^2 + 1/9^2 + 1/10^2 + ... = 0.19268664680716093796587180181377725504571855796690...

Cf. A002808, A275647.

First[RealDigits[Zeta[2] - PrimeZetaP[2] - 1, 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Equals Sum_{k>=1} 1/A002808(k)^2.

Equals A275647 - 1.

A284748 Decimal expansion of the sum of reciprocals of composite powers.

Original entry on oeis.org

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

Offset: 0

Terry D. Grant, Apr 01 2017

			Equals 1/(4*3)+1/(6*5)+1/(8*7)+1/(9*8)+1/(10*9)+...
= 0.226843330950204872135632540144057604...

Cf. A066247, A077761, A179119, A185380.
Decimal expansion of the sum of reciprocal powers: A136141 (primes), A154945 (primes at even powers), A152447 (semiprimes), A154932 (squarefree semiprimes).
Decimal expansion of the 'nonprime zeta function': A275647 (at 2), A278419 (at 3).

RealDigits[ NSum[Zeta[n]-1-PrimeZetaP[n], {n, 2, Infinity}], 10, 105] [[1]]

1 - sumeulerrat(1/(p*(p-1))) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{n>=1} 1/A002808(n)^(n+1) = (A275647 - 1) + (A278419 - 1) + ...
Equals Sum_{n>=1} 1/A002808(n)*(A002808(n)-1).
Equals Sum_{n>=2} (Zeta(n) - PrimeZeta(n) - 1) = Sum_{n>=2} CompositeZeta(n).
Equals 1 - A136141.

More digits from Vaclav Kotesovec, Jan 13 2021

A335590 Decimal expansion of the sum of the reciprocals of the squares of the perfect powers > 1.

Original entry on oeis.org

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

Offset: 0

Jon E. Schoenfield, Jan 26 2021

			Equals 1/4^2 + 1/8^2 + 1/9^2 + 1/16^2 + 1/25^2 + 1/27^2 + 1/32^2 + 1/36^2 + 1/49^2 + 1/64^2 + 1/81^2 + 1/100^2 + ... = 0.10047532720009377586014895164367950389302883992472...

Cf. A001597, A013661, A085548, A275647, A335086, A335589, A340588.

RealDigits[Sum[MoebiusMu[k]*(1 - Zeta[2*k]), {k, 2, 200}], 10, 105][[1]] (* Amiram Eldar, Jan 27 2021 *)

suminf(k=2,moebius(k)*(1-zeta(2*k))) \\ Hugo Pfoertner, Jan 27 2021

Equals Sum_{k>=2} 1/A001597(k)^2.

Equals Sum_{k>=2} mu(k)*(1 - zeta(2*k)). - Amiram Eldar, Jan 27 2021

cp's OEIS Frontend

A062312 Nonprime numbers squared.

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A278419 Decimal expansion of sum of cubes of reciprocals of nonprime numbers.

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

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A335589 Decimal expansion of the sum of the reciprocals of the squares of composite numbers.

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A284748 Decimal expansion of the sum of reciprocals of composite powers.

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A335590 Decimal expansion of the sum of the reciprocals of the squares of the perfect powers > 1.

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PARI

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