A100127 -id:A100127 - cp's OEIS frontend

A346173 Decimal expansion of Sum_{k>=1} prime(k)/2^prime(k).

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

Offset: 1

Amiram Eldar, Jul 08 2021

This constant is irrational (Hančl and Tijdeman, 2004).

Jaroslav Hančl and Robert Tijdeman, On the irrationality of Cantor series, Journal für die reine und angewandte Mathematik, Vol. 2004, No. 571 (2004), pp. 145-158.

Cf. A034785, A098990, A100127.

RealDigits[Sum[Prime[n]/2^Prime[n], {n, 1, 100}], 10, 100][[1]]

suminf(k=1, prime(k)/2^prime(k)) \\ Michel Marcus, Jul 09 2021

1.09306425770250716540258595269676368295547596540121...

A110123 Triangle read by rows: T(n,k) is the number of Delannoy paths of length n, having k EE's and NN's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1 or two consecutive N steps from the line y = x-1 to the line y = x+1).

Original entry on oeis.org

1, 3, 11, 2, 45, 16, 2, 197, 100, 22, 2, 903, 576, 174, 28, 2, 4279, 3206, 1202, 266, 34, 2, 20793, 17568, 7732, 2128, 376, 40, 2, 103049, 95592, 47676, 15452, 3408, 504, 46, 2, 518859, 518720, 286156, 105528, 27500, 5096, 650, 52, 2, 2646723, 2813514

Offset: 0

Emeric Deutsch, Jul 13 2005

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).
Row 0 has one term; row n has n terms (n > 0).
Row sums are the central Delannoy numbers (A001850).
Column 0 yields the little Schroeder numbers (A001003).

			T(2,1)=2 because we have NEEN and ENNE.
Triangle begins:
    1;
    3;
   11,   2;
   45,  16,   2;
  197, 100,  22,   2;

Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001003, A001850, A006318, A100127, A110121.

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=simplify((1-z*R*t+z*R)/(1-z-z*R*t+z^2*R*t-z*R-z^2*R)): Gser:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: 1; for n from 1 to 10 do seq(coeff(t*P[n],t^k),k=1..n) od; # yields sequence in triangular form

Sum_{k=0..n-1} k*T(n,k) = 2*A110127(n).

G.f.: (1 - tzR + zR)/(1 - z - tzR + tz^2*R - zR - z^2*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).

A100128 Decimal expansion of Sum_{n>0} 1/(n*prime(n)^n).

Original entry on oeis.org

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

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004

			0.5583276222317691586142791968036046951752521282087420173902229174235854763316376176520093696726521040...

Cf. A094289, A100124, A100126, A100127.

PARI
```
suminf(k=1, 1/(k*prime(k)^k))
```

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A346173 Decimal expansion of Sum_{k>=1} prime(k)/2^prime(k).

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A110123 Triangle read by rows: T(n,k) is the number of Delannoy paths of length n, having k EE's and NN's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1 or two consecutive N steps from the line y = x-1 to the line y = x+1).

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A100128 Decimal expansion of Sum_{n>0} 1/(n*prime(n)^n).

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