%I #33 Sep 25 2024 09:29:40
%S 0,1,0,2,4,1,-2,3,0,5,10,2,7,-1,4,4,9,1,-7,6,-2,11,24,3,8,8,0,0,-8,5,
%T 5,5,10,10,2,2,15,-6,7,7,20,-1,-22,12,4,25,59,4,-4,9,9,9,22,1,14,1,-7,
%U -7,-28,6,6,6,-2,6,11,11,11,11,24,3,3,3,-18,16
%N a(n) = r in a log_2-like sequence of algebraic integers f(n) = (r+phi*s), where phi = (1+sqrt(5))/2 is the golden ratio.
%C The corresponding s value is A375493(n).
%C f(n) is logarithmic with f(x*y) = f(x) + f(y) and in particular this requires f(1) = 0.
%C The value of f(2) is chosen to be f(2) = 1.
%C For odd primes n, f(n) is a linear interpolation f(p) = ( f(p-1) + phi*f(p+1) ) / (1 + phi), biased towards the higher f(p+1).
%C Since 1/(1 + phi) = 2 - phi, the r and s coefficients are always integers.
%C The two sequences can also be generated by a pair of co-recursive integer functions with no reference to phi.
%C The sequence is not injective and not monotonic.
%F f(n) = a(n) + phi*A375493(n).
%e n = 87654321
%e a(n) = 441
%e A375493(n) = -256
%e f(n) = 441+phi*(-256) = 26.78329888
%e Compare log_2(87654321) = 26.38532187
%o (Python)
%o from sympy import primefactors, isprime
%o def a(n): # the present sequence
%o if n in {1,2}: return n-1
%o if isprime(n): return 2*a(n-1) + b(n+1) - (a(n+1) + b(n-1))
%o return (lambda f: a(f) + a(n//f))(primefactors(n)[0])
%o def b(n): # A375493
%o if n in {1,2}: return 0
%o if isprime(n): return b(n-1) + a(n+1) - a(n-1)
%o return (lambda f: b(f) + b(n//f))(primefactors(n)[0])
%Y Cf. A001622, A375493.
%K sign
%O 1,4
%A _Michael J Norris_, Aug 17 2024