A136141 -id:A136141 - cp's OEIS frontend

A382969 The excess of the n-th noncubefree number.

2, 3, 2, 2, 4, 2, 3, 2, 2, 5, 3, 3, 3, 2, 4, 2, 3, 3, 2, 2, 6, 2, 2, 4, 2, 4, 3, 2, 3, 2, 2, 5, 3, 3, 4, 4, 2, 3, 4, 2, 2, 7, 2, 2, 3, 2, 5, 2, 2, 3, 2, 5, 4, 2, 3, 2, 2, 2, 4, 3, 3, 2, 2, 2, 6, 3, 4, 3, 2, 4, 2, 5, 2, 5, 2, 2, 3, 2, 4, 4, 2, 3, 3, 3, 8, 2, 2

Offset: 1

Amiram Eldar, Apr 10 2025

			a(1) = 2 since the 1st noncubefree number is A046099(1) = 8 = 2^3. It has 3 prime factors when counted with multiplicity, and 1 distinct prime factor, so a(1) = 3 - 1 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A046099, A046660, A136141, A275699, A376366.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[Max[e] < 3, Nothing, Total[e] - Length[e]]]; Array[f, 100]

list(lim) = {my(e); for(k = 2, lim, e = factor(k)[,2]; if(vecmax(e) > 2, print1(vecsum(e) - #e, ", ")));}

a(n) = A046660(A046099(n)).
a(n) >= 2.
Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = ((Sum_{p prime} 1/(p*(p-1))) - (1/zeta(3)) * (Sum_{p prime} (p-1)/(p^3-1))) / (1-1/zeta(3)) = 3.12223294188308957729... .

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

A336908 Decimal expansion of Sum_{p prime} (p^2 + p - 1)/(p^2 *(p - 1)^2).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Aug 07 2020

The asymptotic variance of Omega(k) - omega(k) (A046660).

The asymptotic mean of Omega(k) - omega(k) is Sum_{p prime} 1/(p*(p-1)) = 0.773156... (A136141).

			1.695974243757364917275077225546134160625109953018611...

Persi Diaconis, Frederick Mosteller and Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors - Including the square free case, Journal of Number Theory, Vol. 9, No. 2 (1977), pp. 187-202.
Ali Rejali, On the Asymptotic Expansions for the Moments and the Limiting Distributions of Some Additive Arithmetic Functions, Ph.D. dissertation, Department of Statistics, Stanford University, 1978. See p. 59.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Wikipedia, Prime zeta function.

Cf. A001221, A001222, A046660, A136141.

m = 100; RealDigits[PrimeZetaP[2] + NSum[n * PrimeZetaP[n], {n, 3, Infinity}, WorkingPrecision -> 2*m, NSumTerms -> 3*m], 10, m][[1]]

sumeulerrat((p^2 + p - 1)/(p^2 *(p - 1)^2)) \\ Hugo Pfoertner, Aug 08 2020

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} d(k)^2 - ((1/m) * Sum_{k=1..m} d(k))^2, where d(k) = Omega(k) - omega(k) = A001222(k) - A001221(k) = A046660(k).
Equals P(2) + Sum_{k>=3} k*P(k), where P is the prime zeta function.
Equals A086242 -A085548 +A136141 . - R. J. Mathar, Aug 19 2022

A380688 Decimal expansion of Sum_{p prime} (p + 1)^3/((p - 1)^2*p^3).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 31 2025

			4.1586396688963117979214456647351...

Index to constants which are prime zeta sums {3,2,-3}.

Cf. A085541, A085548, A136141, A152441.

sumeulerrat((p + 1)^3/((p - 1)^2*p^3)) \\ Amiram Eldar, Apr 02 2025

Equals 5*A085548 + A085541 - 12*A136141 + 8*A152441.

Equals P(2) + 5*P(3) + Sum_{k>=4} (8*k - 20) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A380689 Decimal expansion of Sum_{p prime} (p + 1)^3/((p - 1)^3*p^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Mar 31 2025

			7.86666539733180449247691932247069085597893471675885...

Index to constants which are prime zeta sums {2,3,-3}.

Cf. A085548, A086242, A136141, A380840.

sumeulerrat((p + 1)^3/((p - 1)^3*p^2)) \\ Amiram Eldar, Apr 02 2025

Equals -A085548 + 6*A136141 - 4*A086242 + 8*A380840.

Equals P(2) + Sum_{k>=3} (4*k^2 - 16*k + 18) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A382553 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p^2).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.2809990766901105306950615756428028421153486312488377838...

Index to constants which are prime zeta sums {2,2,0}.

Cf. A085548, A086242, A136141.

sumeulerrat(1/((p-1)^2*p^2)) \\ Amiram Eldar, Apr 01 2025

Equals A085548 - 2*A136141 + A086242.

Equals Sum_{k>=4} (k-3) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382557 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.1348524669808244377603507952848150816452507...

Index to constants which are prime zeta sums {3,2,0}.

Cf. A086242, A085541, A085548, A136141.

sumeulerrat(1/((p-1)^2*p^3)) \\ Amiram Eldar, Apr 01 2025

Equals 2*A085548 + A085541 - 3*A136141 + A086242.

Equals Sum_{k>=5} (k-4)*P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382560 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*p).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.545620772059739886880398135615633098083906111...

Index to constants which are prime zeta sums {1,3,0}.

Cf. A086242, A136141, A380840.

sumeulerrat(1/((p-1)^3*p)) \\ Amiram Eldar, Apr 01 2025

Equals A136141 - A086242 + A380840.

Equals Sum_{k>=4} ((k-3)*(k-2)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382561 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*p^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.12976922838880491842498576468801517432330671985244961...

Index to constants which are prime zeta sums {3,3,0}.

Cf. A085541, A085548, A086242, A136141, A380840.

sumeulerrat(1/((p-1)^3*p^3)) \\ Amiram Eldar, Apr 01 2025

Equals -3*A085548 - A085541 + 6*A136141 - 3*A086242 + A380840.

Equals Sum_{k>=6} ((k-5)*(k-4)/2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382565 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*(p + 1)).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.411685848545818051738761053275548060524049791198344603239286000...

Index to constants which are prime zeta sums {0,2,1}.

Cf. A086242, A136141, A179119.

sumeulerrat(1/((p-1)^2*(p+1))) \\ Amiram Eldar, Apr 02 2025

Equals -A136141/4 + A086242/2 - A179119/4.

Equals Sum_{k>=2} (k-1) * (P(2*k-1) + P(2*k)), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

cp's OEIS Frontend

A382969 The excess of the n-th noncubefree number.

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

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A336908 Decimal expansion of Sum_{p prime} (p^2 + p - 1)/(p^2 *(p - 1)^2).

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A380688 Decimal expansion of Sum_{p prime} (p + 1)^3/((p - 1)^2*p^3).

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A380689 Decimal expansion of Sum_{p prime} (p + 1)^3/((p - 1)^3*p^2).

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A382553 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p^2).

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A382557 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p^3).

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A382560 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*p).

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