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

0,2

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.

Links

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

Formula

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

Example

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.

Prog

(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

Crossrefs

Cf. A001008, A002805, A002387.

Sequence in context: A119978 A139234 A039784 * A323678 A028243 A003493

Adjacent sequences: A352023 A352024 A352025 * A352027 A352028 A352029

Keyword

nonn

Author

Sebastian F. Orellana, Feb 28 2022

Extensions

More terms from Peter Luschny, Mar 01 2022

Status

approved