OFFSET
-1,1
LINKS
Math StackExchange, A series related to prime numbers
FORMULA
G.f.: f'(t)/f(t), where f(t) = Sum_{p prime} t^p.
For each prime p, we have: p = 2 + Sum_{q<p, q prime} a(p-1-q).
Conjecture 1: lim_{n->infinity} a(n)/a(n+1) = -0.629233... = gamma (A078756).
Conjecture 2: -0.629233... is a root of f(t), where f(t) = Sum_{p prime} t^p.
Conjecture 3: From conjecture 2 it follows that, -0.629233... is also a root of H(t) = Sum_{n>=1} pi(n)*t^n, where pi = A000720 is the prime counting function. This follows since H(t) = f(t)/(1-t).
Inductive definition of a(n): Let b(n,k) be the number of ordered ways of writing n as a sum of k primes. The numbers a(n) might be computed inductively using: n*b(n,1) = Sum_{v=0..n} a(v-1) * b(n-v,1), from which one sees that a(n) is an integer.
Recurrence relation: b(n,k) = k/(n-2*k) * Sum_{v=0..n-1} b(v,k) * a(n-1-v).
EXAMPLE
We have f'(t)/f(t) = 2*t^(-1) + 1 + (-1)*t + 4*t^2 + (-5)*t^3 + 11*t^4 + (-16)*t^5 + 22*t^6 + (-37)*t^7 + 67*t^8 + (-101)*t^9 + 166*t^10 + (-260)*t^11 + 404*t^12 + (-652)*t^13 ...
so a(-1) = 2 because of 2*t^(-1) and a(0) = 1, a(1) = -1, a(2) = 4, etc.
Relation to primes:
3 = 2 + a(0) = 2+1;
5 = 2 + a(2) + a(1) = 2+4-1;
7 = 2 + a(4) + a(3) + a(1) = 2+11-5-1.
PROG
(Sage)
var('t')
def f(n, t):
return sum([ t**(p) for p in primes(n)])
[c for c, d in (derivative(f(251, t))/f(251, t)).series(t, 50).coefficients()]
(PARI) f(n, x) = sum(k=1, n, x^prime(k));
my(N=50, x = 'x + O('x^N), np = primepi(N)); Vec(deriv(f(N, x))/f(N, x)) \\ Michel Marcus, May 10 2019
CROSSREFS
KEYWORD
sign
AUTHOR
Orges Leka, May 08 2019
STATUS
approved