A309816 a(n) is the 2-adic valuation of A014664(n).

1, 2, 0, 1, 2, 3, 1, 0, 2, 0, 2, 2, 1, 0, 2, 1, 2, 1, 0, 0, 0, 1, 0, 4, 2, 0, 1, 2, 2, 0, 1, 2, 1, 2, 0, 2, 1, 0, 2, 1, 2, 0, 5, 2, 0, 1, 0, 1, 2, 0, 0, 3, 1, 4, 0, 2, 0, 2, 1, 1, 2, 1, 0, 2, 2, 1, 0, 1, 2, 3, 0, 0, 2, 1, 0, 2, 2, 3, 2, 1, 2, 0, 3, 0, 1, 5, 2

Offset: 2

Author

Felix Fröhlich, Aug 18 2019

Keywords

• nonn
• easy

Comments

Let p and q be distinct odd primes. Then there exists an integer i such that 2^i == -1 (mod p*q) if and only if a(u) = a(v) and a(u), a(v) > 0, where u and v are the indices of p and q in A000040, respectively (cf. Anderson, Preece, 2008, Lemma 1.4).

Examples

			For n = 7: A014664(7) = 8 and the 2-adic valuation of 8 is 3, since 2^3 = 8, so a(7) = 3.
		

Links

• I. Anderson and D. A. Preece, A general approach to constructing power-sequence terraces for Z_n, Discrete Mathematics, Vol. 308, No. 5-6 (2008), 631-644.

Crossrefs

Cf. A000040, A014664.

Programs

PARI

a(n) = valuation(znorder(Mod(2, prime(n))), 2);

Python

from sympy import n_order, prime
def A309816(n): return (~(m:=n_order(2,prime(n))) & m-1).bit_length() # Chai Wah Wu, Nov 10 2023