A359147 Partial sums of A002326.

1, 3, 7, 10, 16, 26, 38, 42, 50, 68, 74, 85, 105, 123, 151, 156, 166, 178, 214, 226, 246, 260, 272, 295, 316, 324, 376, 396, 414, 472, 532, 538, 550, 616, 638, 673, 682, 702, 732, 771, 825, 907, 915, 943, 954, 966, 976, 1012, 1060, 1090, 1190, 1241, 1253, 1359, 1395, 1431

Offset

0,2

Comments

a(n)/n is the average order of 2 mod m, averaged over all odd numbers m from 1 to 2n+1. From Kurlberg-Pomerance (2013), this is of order constant*n/log(n). So the graph of this sequence grows like constant*n^2/log(n). [The asymptotic formula involves the constant B = 0.3453720641..., A218342. - Amiram Eldar, Feb 15 2023]

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Pär Kurlberg and Carl Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra & Number Theory, Vol. 7, No. 4 (2013), pp. 981-999.

Formula

a(n) = Sum_{k = 0..n} A007733(2*k+1). - Thomas Scheuerle, Feb 15 2023

Maple

a:= proc(n) option remember;
     `if`(n=0, 1, a(n-1)+numtheory[order](2, 2*n+1))
    end:
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 14 2023

Mathematica

Accumulate[MultiplicativeOrder[2, #]&/@Range[1, 151, 2]] (* Harvey P. Dale, Jul 08 2023 *)

Prog

(PARI) a(n) = sum(k = 0, n, if(k<0, 0, znorder(Mod(2, 2*k+1)))) \\ Thomas Scheuerle, Feb 14 2023
(Python)
from sympy import n_order
def A359147(n): return sum(n_order(2, m) for m in range(1, n+1<<1, 2)) # Chai Wah Wu, Feb 14 2023

Crossrefs

Cf. A002326, A007733, A218342.

Sequence in context: A088636 A114113 A100056 * A163714 A333910 A307191

Adjacent sequences: A359144 A359145 A359146 * A359148 A359149 A359150

Keyword

nonn

Author

N. J. A. Sloane, Feb 14 2023

Status

approved