A168218 -id:A168218 - cp's OEIS frontend

A345928 Decimal expansion of Integral_{x>=0} (zeta(x)-1) dx (negated).

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

Offset: 0

Amiram Eldar, Jun 29 2021

Robinson (1980) conjectured and Newman and Widder (1981) proved that this integral is equal to the limit given in the Formula section.

			-0.2432383428909807554150591354654623071783049...

Murray S. Klamkin (ed.), Problems in Applied Mathematics: Selections from SIAM Review, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1990, pp. 281-282.

H. P. Robinson, A Conjectured Limit, Problem 80-7, SIAM Review, Vol. 22, No. 2 (1980), p. 229; D. J. Newman and D. V. Widder, Solution, ibid., Vol. 23, No. 2 (1981), pp. 256-257.

Cf. A099769, A168218.

Equals lim_{n->oo} (Sum_{k=2..n} 1/log(k) - Integral_{x=0..n} (1/log(x)) dx).

A349522 Decimal expansion of Sum_{k>=2} 1/(k*log(k))^2.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Nov 20 2021

Theorem: Bertrand series Sum_{n>=2} 1/(n^q*log(n)^r) is convergent if q > 1.

Application for q = 2 with A201994 (r=-2), A073002 (r=-1), A013661 (r=0), A168218 (r=1), this sequence (r=2).

			0.6926058...

Wikipédia, Série de Bertrand (in French).
Wikipedia, Joseph Bertrand.

Cf. A013661, A073002, A168218, A201994.

sumpos(k=2, 1/(k*log(k))^2) \\ Michel Marcus, Nov 21 2021

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

Equals Integral_{x>=2, y>=2} (zeta(x + y - 2) - 1) dx dy. - Amiram Eldar, Nov 21 2021

A349641 Decimal expansion of the Sum_{k>=2} 1/(k^3*log(k)).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 24 2021

			Sum_{k>=2} 1/(k^3*log(k)) = 0.23799610019862130199...

Similar sequences: A013661, A002117, A073002, A244115, A168218.

(* following Jean-François Alcover's Mathematica program for A168218 *) digits = 110; NSum[ 1/(n^3*Log[n]), {n, 2, Infinity}, NSumTerms -> 500000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 12}}] // RealDigits[#, 10, digits] & // First

intnum(x=3, [oo, log(3)], zeta(x)-1) \\ following Charles R Greathouse IV's program for A168218

sumpos(k=2, 1/(k^3*log(k))) \\ Michel Marcus, Nov 27 2021

Equals Integral_{s=3..oo} (zeta(s) - 1) ds.

A352473 Decimal expansion of Sum_{k>=2} (log(k!) / k!).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Mar 17 2022

Sum_{k>=2} (log(k) / k) is divergent but here, this series is convergent. If v(k) = log(k!) / k!, we have 0 <= v(k) <= w(k) = k^2/k! with w(k) that is convergent, hence, this positive series is convergent.

			0.8286471276718785080389169468558478918...

J.-M. Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.13.b p. 250.

Cf. A099769, A115563, A168218, A257812.

Maple
```
sum(log(n!)/n!, n=2..infinity);
```

sumpos(k=2, log(k!)/k!) \\ Michel Marcus, Mar 17 2022

Equals Sum_{k>=2} (log(k!) / k!).

cp's OEIS Frontend

A345928 Decimal expansion of Integral_{x>=0} (zeta(x)-1) dx (negated).

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A349522 Decimal expansion of Sum_{k>=2} 1/(k*log(k))^2.

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A349641 Decimal expansion of the Sum_{k>=2} 1/(k^3*log(k)).

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A352473 Decimal expansion of Sum_{k>=2} (log(k!) / k!).

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