A326714 a(n) = binomial(n,2) + (2-adic valuation of n).

0, 2, 3, 8, 10, 16, 21, 31, 36, 46, 55, 68, 78, 92, 105, 124, 136, 154, 171, 192, 210, 232, 253, 279, 300, 326, 351, 380, 406, 436, 465, 501, 528, 562, 595, 632, 666, 704, 741, 783, 820, 862, 903, 948, 990, 1036, 1081, 1132, 1176, 1226, 1275, 1328, 1378

Offset

1,2

Comments

2^a(n) is the smallest integer m >= n such that binomial(m,n) is divisible by 2^binomial(n,2).

2^a(n) is conjectured to be the order of the smallest n-symmetric graph.

Links

Table of n, a(n) for n=1..53.

Sebastian Jeon, Tanya Khovanova, 3-Symmetric Graphs, arXiv:2003.03870 [math.CO], 2020.

Formula

a(n) = A007814(n) + A161680(n).

Example

Binomial(4,2) is 6. In addition, the 2-adic value of 4 is 2, so a(4) = 8.

Mathematica

a[n_] := Binomial[n, 2] + IntegerExponent[n, 2]; Array[a, 60] (* Giovanni Resta, Dec 03 2019 *)

Prog

(Python)
for i in range(1, 70):
    j = i
    res = i*(i-1)//2
    while j%2 == 0:
        res = res + 1
        j = j // 2
    print(str(res), end = ', ')
(Python)
def A326714(n): return (n*(n-1)>>1)+(~n & n-1).bit_length() # Chai Wah Wu, Jul 01 2022

Crossrefs

Cf. A007814, A161680, A329952.

Sequence in context: A295030 A317655 A060697 * A240217 A265224 A093353

Adjacent sequences: A326711 A326712 A326713 * A326715 A326716 A326717

Keyword

nonn

Author

Sebastian Jeon and Tanya Khovanova, Dec 02 2019

Status

approved