A117897 Number of labeled trees on prime numbers of nodes through n-th prime.

1, 4, 129, 16936, 2357964627, 1794518358664, 2862424846028174457, 5483249282630830360396, 39471589603944768518079950019, 3053134546009996125349281528007992109928

Offset: 1

Author

Jonathan Vos Post, May 03 2006

Keywords

• easy
• nonn

Comments

A000178 = Sum_{k=1..n} k^(k-1). A001923 = Sum_{k=1..n} k^k. A061789 = Sum_{k=1..n} prime(k)^prime(k), prime(k) = k-th prime.

First differences a(n+1) - a(n) for n=1,...,9 are A076931(j) at j=3, 5, 7, 11, 13, 17, 19, 23 and 29. - R. J. Mathar, May 01 2007

Examples

			a(1) = number of labeled trees on prime(1) numbers of nodes = number of labeled trees on 2 nodes = A000272(2) = 2^0 = 1.
a(2) = number of labeled trees on prime(1) or prime(2) numbers of nodes = number of labeled trees on 2 or 3 nodes = A000272(2)+A000272(3) = 2^0 + 3^1 = 4.
a(3) = number of labeled trees on prime(1) or prime(2) or prime(3) numbers of nodes = number of labeled trees on 2 or 3 or 5 nodes = A000272(2)+A000272(3)+A000272(5) = 2^0 + 3^1 + 5^3 = 129.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..75

Crossrefs

Cf. A000040, A000178, A000272, A001923, A061789.

Programs

Mathematica

Table[Sum[Prime[k]^(Prime[k] -2), {k,n}], {n,20}] (* G. C. Greubel, Sep 27 2021 *)

Sage

[sum( nth_prime(k)^(nth_prime(k) -2) for k in (1..n)) for n in (1..20)] # G. C. Greubel, Sep 27 2021

Formula

a(n) = Sum_{k=1..n} prime(k)^(prime(k)-2).

a(n) = Sum_{k=1..n} A000272(A000040(k)).