A389559 Indices where the cumulative sum of cos(2k+1)^(2k+1) reaches a record high value.

0, 12, 15, 37796, 37799, 37802, 37803, 38151, 38154, 38157, 38158, 38506, 38509, 38512, 38513, 38861, 38864, 38867, 38868, 39216, 39219, 39222, 39223, 39571, 39574, 39577, 39578, 39926, 39929, 39932, 39933, 40281, 40284, 40287, 40288, 40636, 40639, 40642, 40643

Offset

0,2

Comments

The sequence has consecutive differences of 3, 3, 1, 348 after the 3rd term for as long as the sequence exists. The differences add up to 355 which is the numerator of a significant convergent to Pi.

It appears that the sequence has only 459 terms with the last term being 77918.

Sum_{k=0..77918} cos(2k+1)^(2k+1) = 78.59291259...

Links

Jwalin Bhatt, Table of n, a(n) for n = 0..458

Example

-----------------------------------------
|   k   | Sum_{n=0..k} cos(2k+1)^(2k+1) |
-----------------------------------------
|   0   |           0.5403...           |
|   1   |          -0.4299...           |
|   11  |           0.3501...           |
|   12  |           1.1519...           |
-----------------------------------------

Prog

(Python)
from mpmath import iv
def a_upto(limit):
    sum_, highs = iv.mpf('0'), [0]
    for i in range(1, 1+limit):
        t = 2*i + 1
        sum_ += iv.cos(t)**t
        if sum_ > 0:
            highs.append(i)
            sum_ = 0
    return highs
print(a_upto(40000)) # Jwalin Bhatt, Nov 13 2025

Crossrefs

Cf. A387706, A389560.

Sequence in context: A079322 A167304 A191966 * A333720 A135451 A175784

Adjacent sequences: A389556 A389557 A389558 * A389560 A389561 A389562

Keyword

nonn

Author

Jwalin Bhatt, Oct 08 2025

Status

approved