A337347 a(n) = a(n-2) + a(n-1) if that sum is prime. Otherwise, a(n) = a(n-2) + a(n-1) + prime(m-1) + prime(m-2) + ... + prime(s), where a(-1) = a(0) = prime(0) = 1, m = pi(a(n-2)), and s = max(0, 1, 2, ..., m-1) such that the sum is prime.

2, 3, 5, 11, 19, 37, 73, 193, 337, 1741, 5851, 39157, 50857, 987713, 1292701, 11168887, 21510131, 177872951, 220893209, 932384951, 15511295531, 43482833879, 974160518137, 2539542521143, 12281147701703, 22439317786231, 47001613189621, 899695689045059

Offset

1,1

Comments

This sequence is similar to Fibonacci sequence except that, if a(n) = a(n-1) + a(n-2) is not a prime, the prime numbers < a(n-2) are added to a(n-1)+a(n-2) sequentially starting from the largest prime number < a(n-2) until the sum becomes a prime.

If the prime numbers added to a(n-1)+a(n-2) are limited to the terms < a(n-2) that are already in the sequence including a(0)=1, we would have a new sequence with 9 terms only: 2, 3, 5, 11, 19, 41, 71, 131, 281.

When a(n-1)+a(n-2) is not prime, the last digit of the last term, prime(s), added to a(n-1)+a(n-2) is either "1", "7", or "9" for n up to 35, except prime(s) = 3 for n = 5 and prime(s) = 23 for n=8. This trend has yet to be verified for n > 35.

Although a(0)=1 is an anchor for the sequence, a(0) does not appear in the sequence because the obvious intent is to construct a sequence of primes. - R. J. Mathar, Jun 18 2021

Links

Table of n, a(n) for n=1..28.

Formula

a(n) = a(n-1) + a(n-2) + Sum{i=1..m-s}prime(m-i), where a(-1) = a(0) = prime(0) = 1, m = pi(a(n-2)), and s = max(0, 1, 2, ..., m) such that a(n) is prime.

Example

a(1) = a(0) + a(-1) = 1 + 1 = 2;
a(2) = a(1) + a(0) = 2 + 1 = 3;
a(3) = a(2) + a(1) = 3 + 2 = 5;
a(4) = a(3) + a(2) + prime(pi(a(2))-1) + prime(pi(a(2))-2) = 5 + 3 + prime(1) + prime(0) = 8 + 2 + 1 = 11;
a(5) = a(4) + a(3) + prime(pi(a(3))-1) = 11 + 5 + prime(2) = 16 + 3 = 19;
a(15) = a(14) + a(13) + prime(pi(a(13))-1) + prime(pi(a(13))-2) + prime(pi(a(13))-3) + prime(pi(a(13))-4) + prime(pi(a(13))-5) = 987713 + 50857 + prime(5208) + prime(5207) + prime(5206) + prime(5205) + prime(5204) = 1038570 + 50849 + 50839 + 50833 + 50821 + 50789 = 1292701.

Prog

(Python)
from sympy import isprime, prime, primepi
a_1 = 1
a0 = 1
for n in range(1, 101):
    a = a_1 + a0
    m = primepi(a_1)
    b = 0
    while not isprime(a + b):
        m = m - 1
        b = b + prime(m)
    a = a + b
    print(a)
    a_1 = a0
    a0 = a

Crossrefs

Cf. A000045 (Fibonacci), A000213 (tribonacci).

Sequence in context: A084573 A155954 A358717 * A087581 A360320 A049888

Adjacent sequences: A337344 A337345 A337346 * A337348 A337349 A337350

Keyword

nonn

Author

Ya-Ping Lu, Aug 25 2020

Status

approved