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

Author

Leroy Quet, Jun 12 2007

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A008578.

Programs

Mathematica

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

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

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

Extensions

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