A086242 -id:A086242 - cp's OEIS frontend

A361089 a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.

3, 5, 8, 21, 76, 389, 2679, 23969, 269777, 3717613, 61326301, 1188642478, 26651213526, 682263659097, 19720607003199, 637490095320530, 22857266906194526, 902495758030572213, 38993221443197045348, 1833273720522384358862

Offset: 2

Artur Jasinski, Jun 11 2023

nonn

Because lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)) = 2.7977647035208... (see A363078) then a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.

			a(2) = 3 because Sum_{k=2..3} 1/(k*log(log(k))) = 2.18008755... > 2 and Sum_{k=2..2} 1/(k*log(log(k))) = -1.364208386450... < 2.
a(7) = 389 because Sum_{k=2..389} 1/(k*log(log(k))) = 7.000345... > 7 and Sum_{k=2..388} 1/(k*log(log(k))) = 6.99890560988... < 7.

Pascal Sebah, Table of n, a(n) for n = 2..35

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A363078.

(*slow procedure*)
lim = 2; sum = 0; aa = {}; Do[sum = sum + N[1/(k Log[Log[k]]), 100];
 If[sum >= lim, AppendTo[aa, k]; Print[{lim, sum, k}];
  lim = lim + 1], {k, 2, 269777}];aa
(*quick procedure *)
aa = {3}; cons = 2.79776470352080492766050456553352884330850083202326989577856315;
Do[ww = w /. NSolve[LogIntegral[Log[w]] + cons == n, w];
 AppendTo[aa, Round[ww][[1]]], {n, 3, 21}]; aa

For n >= 3, a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.

A363078 Decimal expansion of lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jun 11 2023

Value computed and communicated by Pascal Sebah.

For the smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n see A361089.

			2.7977647035208...

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A361089.

A091589 Decimal expansion of the analog of the Mertens constant B_2 in the asymptotic series for the variance of the number of prime factors Omega.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 22 2004

			0.76478480...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 95.

Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001222, A013661 (Pi^2/6), A086242, A083342.

Equals A083342 minus Pi^2/6 plus A086242. - R. J. Mathar, Oct 14 2010

Equals lim_{m->oo} ((1/m) * Sum_{k=1..m} A001222(k)^2 - ((1/m) * Sum_{k=1..m} A001222(k))^2 - log(log(m))). - Amiram Eldar, Jan 09 2024

More digits from R. J. Mathar, Oct 14 2010

More digits (using terms in A083342 and A086242) from Vaclav Kotesovec, Aug 12 2019

A382552 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.60190832569884016005288567066649730356085862...

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

Cf. A086242, A136141.

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

Equals -A136141 + A086242.

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

A382584 Decimal expansion of Sum_{p prime} 1/((p - 1)^2p(p + 1)).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.1902224771530221083141246173909492430368088328937868071588972676186916269020795654200305...

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

Cf. A086242, A136141, A179119.

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

Equals -3*A136141/4 + A086242/2 + A179119/4.

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

A119723 Numerator of Sum[ (-1)^(k-1) * 1/(Prime[k]-1)^2, {k,1,n}].

Original entry on oeis.org

1, 3, 13, 113, 2861, 709, 45601, 408809, 49595489, 2426258561, 485934733, 485460413, 2429223061, 2427480661, 1284905668069, 217047437215261, 182605590283392901, 36508279615059377, 36518889897389297

Offset: 1

Alexander Adamchuk, Jun 13 2006

Eric Weisstein's World of Mathematics, Prime Sums.

Cf. A119686, A006093, A000040, A086242.

Numerator[Table[Sum[(-1)^(i-1)*1/(Prime[i]-1)^2, {i, 1, n}], {n, 1, 30}]]

a(n) = numerator[ Sum[ (-1)^(k-1) * 1/(Prime[k]-1)^2, {k,1,n}]].

A154937 Decimal expansion of sum_q 1/(q-1)^2 over the semiprimes q = 4,6,9,10,...

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 17 2009

Semiprime analog of A086242.

			Equals 0.2097883239400194927... = 1/3^2+1/5^2+1/8^2+1/9^2+1/13^2+...

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], Table 11.

Equals Sum_{i>=1} 1/(A001358(i)-1)^2.

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

A366249 Decimal expansion of lim_{x->oo} (Sum_{primes p<=x} 1/(p*log(log(p)))) - log(log(log(x))).

Original entry on oeis.org

2, 9, 3, 8, 3, 2, 9, 0, 1

Offset: 1

Artur Jasinski, Oct 05 2023

Value computed and communicated by Pascal Sebah.

			2.93832901...

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363078, A363368.

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

cp's OEIS Frontend

A361089 a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.

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A363078 Decimal expansion of lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)).

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A091589 Decimal expansion of the analog of the Mertens constant B_2 in the asymptotic series for the variance of the number of prime factors Omega.

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Extensions

A382552 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p).

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PARI

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A382584 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p*(p + 1)).

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A119723 Numerator of Sum[ (-1)^(k-1) * 1/(Prime[k]-1)^2, {k,1,n}].

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A154937 Decimal expansion of sum_q 1/(q-1)^2 over the semiprimes q = 4,6,9,10,...

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

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A366249 Decimal expansion of lim_{x->oo} (Sum_{primes p<=x} 1/(p*log(log(p)))) - log(log(log(x))).

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A382584 Decimal expansion of Sum_{p prime} 1/((p - 1)^2p(p + 1)).