A361199 - OEIS

%I #42 Mar 22 2023 12:51:15

%S 1,2,2,2,2,1,3,2,2,3,7,2,3,7,3,5,3,7,3,7,4,4,1,10,9,2,5,7,6,4,4,5,11,

%T 8,6,2,4,7,15,6,5,10,12,9,7,11,7,14,9,8,7,16,11,9,11,10,8,7,11,13,13,

%U 9,15,9,13,14,7,15,9,12,14,15,5,13,12,6,12,9,15

%N a(1) = 1, a(2) = 2; for n >=3, a(n) is the number of primes in a(n-1), a(n-1) + a(n-2), ..., a(n-1) + a(n-2) + ... + a(1).

%H Michael S. Branicky, <a href="/A361199/b361199.txt">Table of n, a(n) for n = 1..10000</a>

%e a(10) = 3 because three primes result from the adding up process, visualized below. These are 2, 7 and 17.

%e .                                                      Prime P/    Prime

%e a(1)  a(2)  a(3)  a(4)  a(5)  a(6)  a(7)  a(8)  a(9)  Composite C  Count

%e .

%e                                                   2  =   2  P        1

%e                                             2  +  2  =   4  C        0

%e                                       3  +  2  +  2  =   7  P        1

%e                                 1  +  3  +  2  +  2  =   8  C        0

%e                           2  +  1  +  3  +  2  +  2  =  10  C        0

%e                     2  +  2  +  1  +  3  +  2  +  2  =  12  C        0

%e               2  +  2  +  2  +  1  +  3  +  2  +  2  =  14  C        0

%e         2  +  2  +  2  +  2  +  1  +  3  +  2  +  2  =  16  C        0

%e   1  +  2  +  2  +  2  +  2  +  1  +  3  +  2  +  2  =  17  P        1

%e                                                                  + _____

%e                                                          a(10)   =   3

%e .

%p A[1]:= 1: A[2]:= 2: S:= [0,1,3]:

%p for n from 3 to 100 do

%p   A[n]:= nops(select(isprime, map(t -> S[n]-t, S[1..n-1])));

%p   S:= [op(S),A[n]+S[-1]]

%p od:

%p seq(A[i],i=1..100); # _Robert Israel_, Mar 22 2023

%t a[1] = 1; a[2] = 2; a[n_] := a[n] = Count[Accumulate[Table[a[i], {i, n - 1, 1, -1}]], _?PrimeQ]; Array[a, 100] (* _Amiram Eldar_, Mar 04 2023 *)

%o (PARI) lista(nn) = my(va=vector(nn)); va[1] = 1; va[2] = 2; for (n=3, nn, my(s=0, nb=0); for (k=1, n-1, s += va[n-k]; if (isprime(s), nb++);); va[n] = nb;); va; \\ _Michel Marcus_, Mar 04 2023

%o (Python)

%o from sympy import isprime

%o from itertools import islice

%o def agen(): # generator of terms

%o     an, sums = 2, [1]

%o     yield 1

%o     while True:

%o         yield an

%o         sums = [s + an for s in sums] + [an]

%o         an = sum(1 for s in sums if isprime(s))

%o print(list(islice(agen(), 80))) # _Michael S. Branicky_, Mar 22 2023

%Y Cf. A000040, A361231, A361178.

%K nonn

%O 1,2

%A _Tamas Sandor Nagy_, Mar 04 2023