A261819 -id:A261819 - cp's OEIS frontend

A084448 Decimal expansion of (negative of) Kinkelin constant.

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

Offset: 0

N. J. A. Sloane, Jun 27 2003

Named after the Swiss mathematician Hermann Kinkelin (1832-1913). - Amiram Eldar, Jun 16 2021

			-0.1654211437004509292139196602427806427640363803352017836665223...

Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., Vol. 7, No. 4 (1998), pp. 343-359.
Hermann Kinkelin, Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung, J. Reine Angew. Math., Vol. 57 (1860), pp. 122-158; alternative link. See eq. (22), p. 133.
E. M. Wright, Asymptotic partition formulae, I: Plane partitions, Quart. J. Math., Vol. 2 (1931), pp. 177-189.

Cf. A001620, A074962, A084539.

Maple
```
Digits := 200; evalf(Zeta(1,-1));
```

RealDigits[1/12 - Log[Glaisher], 10, 99] // First (* Jean-François Alcover, Feb 15 2013 *)

-zeta'(-1) \\ Charles R Greathouse IV, Dec 12 2013

Zeta(1, -1). Almkvist gives many formulas.
Equals (1 - gamma - log(2*Pi))/12 + Zeta'(2)/(2*Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 25 2015
From Amiram Eldar, Jun 16 2021: (Start)
Equals 1/24 - gamma/3 - Sum_{k>=1} (zeta(2*k+1)-1)/((2*k+1)*(2*k+3)) = 1/12 - log(A), where A is the Glaisher-Kinkelin constant (A074962) (Kinkelin, 1860).
Equals 2 * Integral_{x>=0} x*log(x)/(exp(2*Pi*x)-1) dx = 2*A261819. (Wright, 1931). (End)

A262531 Minimal nested palindromic primes with seed 000.

Original entry on oeis.org

1300031, 10130003101, 171013000310171, 91710130003101719, 3917101300031017193, 16391710130003101719361, 10116391710130003101719361101, 191011639171013000310171936110191, 91910116391710130003101719361101919, 119191011639171013000310171936110191911

Offset: 2

Clark Kimberling, Sep 24 2015

Let s be a palindrome and put a(1) = s. Let a(2) be the least palindromic prime having s in the middle; for n > 2, let a(n) be the least palindromic prime having a(n-1) in the middle. Then (a(n)) is the sequence of minimal nested palindromic primes with seed s. (For A261819, the seed is not an integer, so that the offset is 2.)

			As a triangle:
          000
        1300031
      10130003101
    171013000310171
   91710130003101719
  3917101300031017193
16391710130003101719361

Clark Kimberling, Table of n, a(n) for n = 2..200

Cf. A261881.

s0 = "000"; s = {ToExpression[s0]};Do[NestWhile[# + 1 &, 1, ! PrimeQ[tmp = FromDigits[Join[#, IntegerDigits[Last[s], 10, Max[StringLength[s0],Length[IntegerDigits[Last[s]]]]], Reverse[#]]&[IntegerDigits[#]]]] &]; AppendTo[s, tmp], {10}]; s0 <> ", " <> StringTake[ToString[Rest[s]], {2, -2}]
(* Peter J. C. Moses, Sep 23 2015 *)

cp's OEIS Frontend

A084448 Decimal expansion of (negative of) Kinkelin constant.

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