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

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

Author

Terry D. Grant, Apr 01 2017

Keywords

• nonn
• cons

Examples

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

Crossrefs

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).

Programs

Mathematica

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

PARI

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

Formula

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.

Extensions

More digits from Vaclav Kotesovec, Jan 13 2021