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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 14 2023
STATUS
approved