A350404 -id:A350404 - cp's OEIS frontend

A350695 Number of solutions to +-2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = n.

1, 0, 1, 0, 1, 0, 1, 1, 4, 5, 9, 15, 26, 45, 77, 137, 243, 434, 774, 1408, 2554, 4667, 8627, 15927, 29559, 54867, 101688, 189425, 355315, 668598, 1264180, 2395462, 4506221, 8507311, 16084405, 30545142, 57898862, 110199367, 209957460, 400430494, 765333684

Offset: 0

Ilya Gutkovskiy, Jan 29 2022

nonn

Cf. A000040, A022894, A058377, A063890, A113040, A261061, A350404.

Table[SeriesCoefficient[Product[x^Prime[k] + 1/x^Prime[k], {k, n - 1}], {x, 0, n}], {n, 0, 40}] (* Stefano Spezia, Jan 30 2022 *)

from sympy import sieve, primerange
from functools import cache
@cache
def b(n, i):
    maxsum = 0 if i < 2 else sum(p for p in primerange(2, sieve[i-1]+1))
    if n > maxsum: return 0
    if i < 2: return 1
    return b(n+sieve[i-1], i-1) + b(abs(n-sieve[i-1]), i-1)
def a(n): return b(n, n)
print([a(n) for n in range(41)]) # Michael S. Branicky, Jan 29 2022

a(n) = [x^n] Product_{k=1..n-1} (x^prime(k) + 1/x^prime(k)).

A369560 a(n) = [x^n] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 16, 38, 91, 225, 547, 1407, 3570, 9250, 24578, 65740, 175626, 470084, 1279101, 3482419, 9547953, 26445796, 73251187, 203818706, 567543095, 1577629707, 4408095456, 12400615844, 34995570604, 99241500366, 282037360250, 795846583187

Offset: 0

Ilya Gutkovskiy, Jan 25 2024

nonn

a(n) is the number of solutions to n = Sum_{i=1..n} c_i * prime(i) with c_i in {-1,0,1}.

Cf. A000040, A022894, A113040, A316706, A350404, A350695, A350880, A369390.

s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 25 2024

Table[Coefficient[Product[x^Prime[k] + 1 + 1/x^Prime[k], {k, 1, n}], x, n], {n, 0, 31}]

A367088 Number of solutions to +- 1 +- 2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = 0 or 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 6, 7, 12, 19, 32, 53, 90, 156, 276, 493, 878, 1566, 2834, 5146, 9396, 17358, 32042, 59434, 110292, 204332, 380548, 713601, 1342448, 2538012, 4808578, 9043605, 17070234, 32268611, 61271738, 116123939, 220993892, 421000142, 802844420, 1534312896

Offset: 0

Ilya Gutkovskiy, Jan 26 2024

nonn

Cf. A008578, A025591, A022894, A022895, A113040, A350404.

cp's OEIS Frontend

A350695 Number of solutions to +-2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = n.

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A369560 a(n) = [x^n] Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

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A367088 Number of solutions to +- 1 +- 2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = 0 or 1.

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