A089612 a(n) = ((-1)^(n+1)*A002425(n)) modulo 5.
1, 4, 1, 3, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 2, 1, 4, 1, 3, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 4, 1, 4, 1, 3, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 2, 1, 4, 1, 3, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 2, 1, 4, 1, 3, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 4, 1, 4, 1, 3, 1, 4, 1, 1, 1
Offset: 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1024
Programs
-
Magma
[Numerator(2/n*(4^n-1)*Bernoulli(2*n)) mod 5: n in [1..100]]; // Vincenzo Librandi, Aug 01 2018
-
Mathematica
Table[Mod[Numerator[2 / n (4^n - 1) BernoulliB[2 n]], 5], {n, 100}] (* Vincenzo Librandi, Aug 01 2018 *)
-
PARI
a(n)=numerator(2/n*(4^n-1)*bernfrac(2*n))%5
-
PARI
a(n)=if(n%2, 1, 2*2^valuation(n,2) % 5); \\ Andrew Howroyd, Aug 01 2018
Formula
Let S(1) = {1, 4} and S(n+1) = S(n)*S'(n), where S'(n) is obtained from S(n) by changing last term using the cyclic permutation 4->3->1->2->4; sequence is S(infinity).
Multiplicative with a(2^e) = 2^(e + 1) mod 5, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Aug 01 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 21/10. - Amiram Eldar, Nov 10 2022