A104589 a(1)=1. a(n) = a(n-1) + (sum of terms, from among terms a(1) through a(n-1), which are prime or 1).

1, 2, 5, 13, 34, 55, 76, 97, 215, 333, 451, 569, 1256, 1943, 2630, 3317, 4004, 4691, 10069, 25516, 40963, 56410, 71857, 87304, 102751, 118198, 133645, 149092, 164539, 179986, 195433, 210880, 226327, 241774, 257221, 529889, 802557, 1075225

Offset

1,2

Comments

By Dirichlet's Theorem there are an infinite number of primes in this sequence.

Links

Michel Marcus, Table of n, a(n) for n = 1..2000

Formula

a(n) = 3*a(n-1) - a(n-2) if a(n-1) is prime, else a(n) = 2*a(n-1) - a(n-2) for n>3. - John Tyler Rascoe, Jul 20 2022

Example

The noncomposites among the first 8 terms of the sequence are 1, 2, 5, 13 and 97. The sum of these is 1+2+5+13+97 = 118. So a(9) = a(8) + 118 = 215.

Mathematica

f[lst_] := Append[lst, Last@ lst + Plus @@ Select[lst, (PrimeQ@ # || # == 1) &]]; Nest[f, {1}, 38] (* Robert G. Wilson v, Jul 02 2007 *)

Prog

(PARI) lista(nn) = my(va = vector(nn), s = 1); va[1] = 1; for (n=2, nn, va[n] = va[n-1] + s; if (isprime(va[n]), s += va[n]); ); va; \\ Michel Marcus, Jul 21 2022
(Python)
from sympy import isprime
from itertools import islice
def A104589_gen(): # generator of terms
    a, b = 1, 1
    while True:
        yield a
        a += b
        b += a if isprime(a) else 0
A104589_list = list(islice(A104589_gen(), 50)) # Chai Wah Wu, Jun 03 2024

Crossrefs

Cf. A008578.

Sequence in context: A271940 A273721 A112841 * A154101 A122024 A252932

Adjacent sequences: A104586 A104587 A104588 * A104590 A104591 A104592

Keyword

nonn

Author

Leroy Quet, Jun 12 2007

Extensions

More terms from Robert G. Wilson v, Jul 02 2007

Status

approved