A352026 a(n) is the nearest integer to 1/(H(k) - n), where H(k) is the smallest harmonic number that exceeds n.

1, 2, 12, 50, 37, 483, 229, 785, 2059, 4806, 23251, 56470, 327690, 813351, 734186, 2643630, 10476269, 67340402, 268822185, 102740092, 618260119, 2491694355, 7222972533, 50525424196, 44010188391, 164490666033, 131444704333, 548839044705, 1808874061272, 9913711133738

Offset: 0

Author

Sebastian F. Orellana, Feb 28 2022

Keywords

• nonn

Comments

The k-th harmonic number, H(k), is Sum_{j=1..k} 1/j. A002387(n) is the smallest k such that H(k) > n.

Examples

			H(2) = 1/1 + 1/2 = 3/2 = 1.5;
H(3) = 1/1 + 1/2 + 1/3 = 11/6 = 1.8333...;
H(4) = 1/1 + 1/2 + 1/3 + 1/4 = 25/12 = 2.08333..., which is the first harmonic number that exceeds 2, so a(2) = round(1/(25/12 - 2)) = round(1/(1/12)) = 12.
H(10) = 7381/2520 = 2.92896...;
H(11) = 83711/27720 = 3.01987..., which is the first harmonic number > 3, and the fractional part of 83711/27720 = 551/27720, so a(3) = round(27720/551) = round(50.30852...) = 50.
		

Links

• Peter Luschny, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A001008, A002805, A002387.

Programs

PARI

a(n)={my(s=0,k=0); while(s<=n, k++;s+=1/k); round(1/(s-n))} \\ Andrew Howroyd, Mar 01 2022

SageMath

RR = RealField(1000)
def A352026(n):
    g = RR.euler_constant()
    u = exp(RR(n) - g)
    a = u + RR(3/2) - RR(1/(24*u)) + RR(3/(640*u^3))
    h = RR(psi(RR(floor(a)))) + g
    return round(RR(1/(h - RR(n))))
print([A352026(n) for n in range(30)])  # Peter Luschny, Mar 03 2022

Formula

a(n) = round(1/(H(A002387(n)) - n)).

Extensions

More terms from Peter Luschny, Mar 01 2022