New hyperoperation and new Recaman sequence

[sweety439]

The hyperoperation is:

a[1]b = a+b
a[2]b = a*b = a+a+a+...+a+a+a with b a's
a[3]b = a^b = a*a*a*...*a*a*a with b a's
a[4]b = a^a^a^...^a^a^a with b a's
a[5]b = a[4]a[4]a[4]...[4]a[4]a[4]a with b a's
a[6]b = a[5]a[5]a[5]...[5]a[5]a[5]a with b a's

etc.

now we define:

a[0]b = max(a,b) (i.e. if a>b, then a[0]b = a, if a<b, then a[0]b = b, if a=b, then a[0]b = a = b)

and

a[-1]b = a-b
a[-2]b = a/b
a[-3]b = log_b(a)
a[-4]b = slog_b(a)
etc.

(i.e. for positive integer n, a[-n]b = c if and only if b[n]c = a)

and for the Recaman sequence, A005132, there are many versions, including A064389, A274647, A274648, A064387, A064388, A350578, A277558, A335299, A171884, now we consider the version A113880: we set a(0) = 0 and a(n) = a(n-1)[k]n with the smallest integer k (skipped k=0, or a(5) will be 2[0]5 = 5 instead of 7) such that a(n-1)[k]n is nonnegative integer and not already in the sequence, it is conjectured that its result is the same as A113880, i.e. k<-2 will never produce a nonnegative integer a(n-1)[k]n, and k>2 is never needed