A200265 Triangle read by rows: coefficients in an asymptotic expansion of the n-th prime.

1, 1, 2, 1, 6, 11, 2, 21, 84, 131, 6, 92, 588, 1908, 2666, 24, 490, 4380, 22020, 62860, 81534, 120, 3084, 35790, 246480, 1075020, 2823180, 3478014, 720, 22428, 322224, 2838570, 16775640, 66811920, 165838848, 196993194

Offset

0,3

Comments

The asymptotic expansion of p_n, the n-th prime, is p_n ~ n*log n + n*(log log n -1) + n * Sum_{k>=1} P_k(log log n)*(log n)^(-k),

where P_k(y) = ((-1)^(k+1)/ k!) Sum_{r=0..k} (-1)^r a_{k,r} y^(k-r), and

where a_{r,k} are natural numbers.

References

M. Cipolla, "La determinazione asintotica dell'n-mo numero primo.", Rend. d. R. Acc. di sc. fis. e mat. di Napoli, s. 3, VIII (1902), pp. 132-166.

Links

Juan Arias-de-Reyna, Table of n, a(n) for n = 0..1484

J. Arias-de-Reyna, J. Toulisse, The n-th prime asymptotically, arXiv:1203.5413 [math.NT], 2012.

P. Dusart, The kth prime is greater than k(ln k+ ln ln k+1) for k>=2, Math. Comp 68 (225) (1999) 411-415 [MR1620223]

G. Mincu, An asymptotic expansion J. Ineq. Pure Appl Math. 4 (2) (2003) #30 [MR1994243]

B. Salvy, Fast computation of some asymptotic functional inverses, J. Symbolic Comput., 17 (1994), 227-236.

N. K. Sinha, On the asymptotic expansion of the sum of the first n primes, arXiv:1011.1667 [math.NT], 2010-2015.

Formula

The polynomials P_k are defined by

P_1(y) = y-2, P_2(y) = -(y^2-6y+11)/2, P_n = n*P_{n-1} - P'_{n-1} + (1/n)*Sum_{k=1..n-1} k*((k-1)*P_{k-1} - P_k - P'_{k-1})*P_{n-k-1}, where P_0(y) = y-1.

Example

Triangle begins
   1;
   1,   2;
   1,   6,   11;
   2,  21,   84,   131;
   6,  92,  588,  1908,  2666;
  24, 490, 4380, 22020, 62860, 81534;
  ...
See a(n,k) table on top of page 14 of article by Arias-de-Reyna and Toulisse.

Mathematica

a[0, -1] = a[0, 0] = a[1, 0] = 1; a[1, 1] = 2; a[n_, k_] /; k == n-1 := a[n, k] = n*a[n-1, k-1] + n*(n-1)*a[n-1, n-1]; a[n_, n_] := a[n, n] = a[n-1, n-1]*n^2 + a[n-1, n-2]*n - (n-1)*Sum[Binomial[n-2, k-1]*(k*a[k-1, k-2] + k*(k-1)*a[k-1, k-1] - a[k, k])*a[n-k-1, n-k-1], {k, 1, n-1}]; a[n_, k_] /; k<0 || k>n = 0; a[n_, k_] := a[n, k] = n*a[n-1, k-1] + (n*(n-1)*a[n-1, k])/(n-k); Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 31 2015 *)

Crossrefs

Sequence in context: A173773 A305512 A121927 * A039762 A039795 A283746

Adjacent sequences: A200262 A200263 A200264 * A200266 A200267 A200268

Keyword

nonn,tabl

Author

Juan Arias-de-Reyna, Nov 15 2011

Status

approved