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A307977
Expansion of g.f.: f'(t)/f(t), where f(t) = Sum_{p prime} t^p.
1
2, 1, -1, 4, -5, 11, -16, 22, -37, 67, -101, 166, -260, 404, -652, 1049, -1653, 2636, -4183, 6632, -10565, 16804, -26676, 42390, -67372, 107061, -170184, 270472, -429784, 683067, -1085561, 1725182, -2741781, 4357291, -6924696, 11005082, -17489663, 27795141, -44173120
OFFSET
-1,1
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
Real root of f(t) is conjectured to be A078756. The prime numbers (A000040) can be recovered from this sequence. It is conjectured that the decimal expansion in A078756 is a root of H(t).
Sequence in context: A270952 A143392 A090668 * A355334 A295259 A255009
KEYWORD
sign
AUTHOR
Orges Leka, May 08 2019
STATUS
approved