A119746 -id:A119746 - cp's OEIS frontend

A381149 a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + sum of prior prime terms.

2, 3, 8, 13, 31, 80, 129, 178, 227, 503, 1282, 2061, 2840, 3619, 4398, 5177, 5956, 6735, 7514, 8293, 17365, 26437, 61946, 97455, 132964, 168473, 203982, 239491, 275000, 310509, 346018, 381527, 798563, 1215599, 1632635, 2049671, 2466707, 5350450, 8234193, 11117936

Offset: 1

James C. McMahon, Feb 15 2025

nonn

			For n=5, a(5) = a(4) + sum of prior primes = 13 + (2 + 3 + 13) = 31, so that 13 is counted twice.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A101135, A119746, A131093, A381150.

Nest[Append[#,#[[-1]]+Total[Select[#,PrimeQ]]]&,{2,3},38]

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [2, 3]
    primesum, an = 5, 3
    while True:
        an += primesum
        if isprime(an): primesum += an
        yield an
print(list(islice(agen(), 40))) # Michael S. Branicky, Feb 19 2025

A381150 a(0) = 1, a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + (sum of prior prime terms or whose negatives are prime) - (sum of prior composite terms or whose negatives are composite).

Original entry on oeis.org

1, 2, 3, 8, 5, 7, 16, 9, -7, -30, -23, -39, -16, 23, 85, 62, -23, -131, -370, -239, -347, -802, -455, 347, 1496, 1149, -347, -2190, -1843, 347, 2884, 2537, -347, -3578, -3231, 347, 4272, 3925, -347, -4966, -4619, 347, 5660, 5313, -347, -6354, -6007, -11667, -5660

Offset: 0

James C. McMahon, Feb 15 2025

sign

			For n=5, a(5) = 5 + (2 + 3 + 5) - 8 = 7.
For n=9, a(9) = -7 + (2 + 3 + 5 + 7 -7) - (8 + 16 + 9) = -7 + 10 - 33 = -30

Michael De Vlieger, Table of n, a(n) for n = 0..10003 (a(n) for n = 1..1000 from James C. McMahon)
Michael De Vlieger, Scatterplot of m*log_10(m*a(n)), n = 1..2^10, where m = 1 of a(n) > 0 (shown in green) and m = -1 if a(n) < 0 (shown in red).
Michael De Vlieger, Scatterplot of m*log_10(m*a(n)), n = 1..2^16, where m = 1 of a(n) > 0 (shown in green) and m = -1 if a(n) < 0 (shown in red).

Cf. A000040, A002808, A101135, A119746, A131093, A381149.

b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+(t->
     `if`(isprime(abs(t)), t, `if`(abs(t)>1, -t, 0)))(a(n)))
    end:
a:= proc(n) option remember; `if`(n<3, n+1, a(n-1)+b(n-1)) end:
seq(a(n), n=0..48);  # Alois P. Heinz, Feb 15 2025

Nest[Append[#,#[[-1]]+Total[Select[#,PrimeQ]]-Total[Select[#,CompositeQ]]]&,{1,2,3},46]

cp's OEIS Frontend

A381149 a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + sum of prior prime terms.

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