A376930 a(0)=0, a(1)=1; for n>1, a(n) = a(n-1)+a(n-2), except where a(n-1) is a prime greater than 2, in which case a(n) = a(n-1)-a(n-2).

0, 1, 1, 2, 3, 1, 4, 5, 1, 6, 7, 1, 8, 9, 17, 8, 25, 33, 58, 91, 149, 58, 207, 265, 472, 737, 1209, 1946, 3155, 5101, 1946, 7047, 8993, 16040, 25033, 8993, 34026, 43019, 8993, 52012, 61005, 113017, 52012, 165029, 217041, 382070, 599111, 981181, 1580292, 2561473

Offset: 0

Author

Stuart Coe, Oct 11 2024

Keywords

• nonn

Comments

It is not clear whether this sequence continues to grow or whether it become stuck in a loop (which could happen if two primes occur in terms n and n-1 or terms n and n-2). Indeed, the sequence is stuck in a loop from around n=10 if we do not ignore the prime number 2.

Similarly, it is not known if the sequence contains any negative terms (which may happen if two primes are adjacent or separated by one other term).

If it continues to grow, it is not clear whether this sequence will contain an infinite number of prime numbers.

Beyond the trivial case of 1, it is not clear if any number will appear more than three times in the sequence. 8993 appears three times, due to several prime terms in close succession.

Also 4618239875200356592 appears three times, as a(111), a(114) and a(117). - Robert Israel, Nov 12 2024

Examples

			a(2) = a(1) + a(0) [as a(1) is not a prime > 2] = 1 + 0 = 1.
a(3) = a(2) + a(1) [as a(2) is not a prime > 2] = 1 + 1 = 2.
a(4) = a(3) + a(2) [as a(3) is not a prime > 2] = 2 + 1 = 3.
a(5) = a(4) - a(3) [as a(4) is a prime > 2]     = 3 - 2 = 1.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..4810

Crossrefs

Cf. A092942, A229137,

Programs

Maple

f:= proc(n) option remember;
      if procname(n-1) > 2 and isprime(procname(n-1)) then procname(n-1) - procname(n-2)
      else procname(n-1) + procname(n-2)
      fi
end proc:
f(0):= 0: f(1):= 1:
seq(f(i),i=0..100); # Robert Israel, Nov 12 2024

Mathematica

s={0,1};Do[If[PrimeQ[s[[-1]]]&&s[[-1]]>2,AppendTo[s,s[[-1]]-s[[-2]]],AppendTo[s,s[[-1]]+s[[-2]]]  ],{n,48}];s (* James C. McMahon, Nov 07 2024 *)

Python

from sympy import isprime
from itertools import islice
def agen(): # generator of terms
    a = [0, 1]
    yield from a
    while True:
        an = a[-1]+a[-2] if a[-1] < 3 or not isprime(a[-1]) else a[-1]-a[-2]
        yield an
        a = [a[-1], an]
print(list(islice(agen(), 50))) # Michael S. Branicky, Oct 11 2024