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

Author

N. J. A. Sloane, Feb 14 2023

Keywords

• nonn

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.

Crossrefs

Cf. A002326, A007733, A218342.

Programs

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 *)

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

Formula

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