A022894 -id:A022894 - cp's OEIS frontend

A350404 Number of solutions to +-2 +- 3 +- 5 +- 7 +- ... +- prime(n) = 0 or 1.

1, 0, 1, 2, 1, 2, 3, 4, 6, 10, 16, 26, 45, 78, 138, 244, 439, 784, 1417, 2572, 4698, 8682, 16021, 29720, 55146, 102170, 190274, 356804, 671224, 1269022, 2404289, 4521836, 8535117, 16134474, 30635869, 58062404, 110496946, 210500898, 401422210, 767158570, 1467402238

Offset: 0

Ilya Gutkovskiy, Dec 29 2021

nonn

			a(6) = 3: 2 + 3 + 5 - 7 + 11 - 13 =
         -2 + 3 + 5 - 7 - 11 + 13 =
         -2 + 3 - 5 + 7 + 11 - 13 = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000040, A022894, A025591, A113040.

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+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:=n-> b(0, n)+b(1, n):
seq(a(n), n=0..45);  # Alois P. Heinz, Jan 16 2022

s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n - 1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1,
     b[n + Prime[i], i - 1] + b[Abs[n - Prime[i]], i - 1]]];
a[n_] := b[0, n] + b[1, n];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

from itertools import product
from sympy import prime, primerange
def a(n):
    if n == 0: return 1
    nn = ["0"] + [str(i) for i in primerange(2, prime(n)+1)]
    return sum(eval("".join([*sum(zip(nn, ops+("", )), ())])) in {0, 1} for ops in product("+-", repeat=n))
print([a(n) for n in range(18)]) # Michael S. Branicky, Jan 16 2022

from sympy import sieve, primerange
from functools import cache
@cache
def b(n, i):
    maxsum = 0 if i == 0 else sum(p for p in primerange(2, sieve[i]+1))
    if n > maxsum: return 0
    if i == 0: return 1
    return b(n+sieve[i], i-1) + b(abs(n-sieve[i]), i-1)
def a(n): return b(0, n) + b(1, n)
print([a(n) for n in range(43)]) # Michael S. Branicky, Jan 16 2022

a(39)-a(40) from Michael S. Branicky, Jan 16 2022

A022900 Number of solutions to c(1)prime(3) + ... + c(n)prime(n+2) = 0, where c(i) = +-1 for i>1, c(1) = 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 8, 0, 22, 0, 42, 0, 147, 0, 663, 0, 1803, 0, 7410, 0, 22463, 0, 87397, 0, 291211, 0, 1091736, 0, 3896012, 0, 13992225, 0, 49681944, 0, 184771042, 0, 677854904, 0, 2495656379, 0, 9260633829, 0, 34281074654, 0, 127420198855, 0

Offset: 1

Clark Kimberling

nonn

			a(8) counts the unique solution {5, -7, 11, -13, 17, -19, -23, 29}.

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A022894 - A022904, A022920.

Cf. A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060, A261045 (r.h.s. = -2).

{f, s} = {3, 0}; Table[t = Map[Prime[# + f - 1] &, Range[2, z]]; Count[Map[Apply[Plus, #] &, Map[t # &, Tuples[{-1, 1}, Length[t]]]], s - Prime[f]], {z, 22}]
(* A022900, a(n) = number of solutions of "sum = s" using Prime(f) to Prime(f+n-1) *)
n = 8; t = Map[Prime[# + f - 1] &, Range[n]]; Map[#[[2]] &, Select[Map[{Apply[Plus, #], #} &, Map[t # &, Map[Prepend[#, 1] &, Tuples[{-1, 1}, Length[t] - 1]]]], #[[1]] == s &]]  (* the unique solution of using n=8 primes; Peter J. C. Moses, Oct 01 2013 *)

A022900(n,rhs=0,firstprime=3)={rhs-=prime(firstprime);my(p=vector(n-1,i,prime(i+firstprime)));sum(i=1,2^(n-1),sum(j=1,#p,(1-bittest(i,j-1)<<1)*p[j])==rhs)} \\ For illustrative purpose, too slow for n >> 20. - M. F. Hasler, Aug 08 2015

a(n) = [x^5] Product_{k=4..n+2} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 28 2024

Corrected and extended by Clark Kimberling, Oct 01 2013
a(23)-a(49) from Alois P. Heinz, Aug 06 2015
Missing cross-references added by M. F. Hasler, Aug 08 2015

A261062 Number of solutions to c(1)prime(2) + ... + c(2n-1)prime(2n) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

0, 0, 1, 0, 6, 8, 30, 121, 385, 1102, 4207, 13263, 48904, 164298, 610450, 2108897, 7592564, 27444148, 100851443, 365507140, 1344593522, 4960584613, 18435632285, 68320148701, 254166868115, 951593812462, 3568369245595, 13386056545363, 50416752718382

Offset: 1

M. F. Hasler, Aug 08 2015

nonn

There cannot be a solution for an even number of terms on the l.h.s. because all terms are odd but the r.h.s. is odd, too.

			a(1) = a(2) = 0 because prime(2) and prime(2) +- prime(3) +- prime(4) are always different from -1.
a(3) = 1 because the solution prime(2) + prime(3) - prime(4) + prime(5) - prime(6) = -1 is the only one involving prime(2) through prime(6).

Alois P. Heinz, Table of n, a(n) for n = 1..300

Cf. A261061, A261063 and A261044 (starting with prime(1), prime(3) and prime(4)), A022894, ..., A022904, A022920, A083309 (r.h.s. = 0, 1 or 2), A261057, A261059, A261060, A261045 (r.h.s. = -2).

s:= proc(n) option remember;
      `if`(n<3, 0, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=2, 1,
      b(abs(n-ithprime(i)), i-1)+b(n+ithprime(i), i-1)))
    end:
a:= n-> b(4, 2*n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 08 2015

s[n_] := s[n] = If[n < 3, 0, Prime[n] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 2, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]];
a[n_] := b[4, 2n];
Array[a, 30] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

A261062(n,rhs=-1,firstprime=2)={rhs-=prime(firstprime);my(p=vector(2*n-2+bittest(rhs,0),i,prime(i+firstprime)));sum(i=1,2^#p-1,sum(j=1,#p,(-1)^bittest(i,j-1)*p[j])==rhs)} \\ For illustrative purpose; too slow for n >> 10.

a(n) = [x^4] Product_{k=3..2*n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024

a(14)-a(29) from Alois P. Heinz, Aug 08 2015

A292698 Number of solutions to 3 +- 7 +- 13 +- 19 +- ... +- prime(4*n) = 0.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 27, 83, 292, 944, 3279, 11291, 38992, 138066, 490248, 1757360, 6347321, 22998089, 83780199, 306819363, 1128999790, 4174251748, 15507225620, 57767819903, 215327188611, 803901214851, 3013081897103, 11331883386143, 42737620941612

Offset: 1

Seiichi Manyama, Sep 21 2017

nonn

			For n = 4 the solution is 3 - 7 - 13 + 19 - 29 + 37 + 43 - 53 = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..200

Cf. A000040, A022894, A083309, A292699, A292700.

s:= proc(n) s(n):= `if`(n=0, 0, ithprime(2*n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      (p-> b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(2*i))))
    end:
a:= n-> b(0, 2*n)/2:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 21 2017

s[n_] := s[n] = If[n == 0, 0, Prime[2n] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1,
     With[{p = Prime[2i]}, b[n+p, i-1] + b[Abs[n-p], i-1]]]];
a[n_] := b[0, 2n]/2;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)

{a(n) = 1/2*polcoeff(prod(k=1, 2*n, x^prime(2*k)+1/x^prime(2*k)), 0)}

Constant term in the expansion of 1/2 * Product_{k=1..2*n} (x^prime(2*k) + 1/x^prime(2*k)).

A292699 Number of solutions to 5 +- 11 +- 17 +- 23 +- ... +- prime(4*n+1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 28, 102, 242, 835, 3381, 10115, 39415, 138555, 465778, 1741167, 6081563, 22156403, 81090107, 301185788, 1098440558, 4071519963, 15235221189, 56764430821, 211797564907, 790029575790, 2962705350767, 11154931819979, 42097079364709

Offset: 1

Seiichi Manyama, Sep 21 2017

nonn

			For n=2 the solution is 5-11-17+23 = 0.
For n=3 the solution is 5+11+17-23+31-41 = 0.
For n=4 the 2 solutions are 5-11+17+23+31+41-47-59 = 0 and 5+11-17+23+31-41+47-59 = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..200

Cf. A000040, A022894, A083309, A292698, A292700.

s:= proc(n) s(n):= `if`(n=0, 0, ithprime(2*n+1)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      (p-> b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(2*i+1))))
    end:
a:= n-> b(0, 2*n)/2:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 21 2017

s[n_] := s[n] = If[n == 0, 0, Prime[2n+1] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1,
     With[{p = Prime[2i+1]}, b[n+p, i-1] + b[Abs[n-p], i-1]]]];
a[n_] := b[0, 2n]/2;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)

{a(n) = 1/2*polcoeff(prod(k=1, 2*n, x^prime(2*k+1)+1/x^prime(2*k+1)), 0)}

Constant term in the expansion of 1/2 * Product_{k=1..2*n} (x^prime(2*k+1) + 1/x^prime(2*k+1)).

A215036 2 followed by "1,0" repeated.

Original entry on oeis.org

2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

N. J. A. Sloane, Aug 06 2012

Take the first n primes and combine them with coefficients +1 and -1; then a(n) is the smallest number (in absolute value) that can be obtained.
For example, a(1) = 2, a(2) = 1 from 3-2 = 1; a(3) = 0 from -2-3+5 = 0; a(11) = 0 from 2-3-5-7+11-13+17+19-23-29+31 = 0.
Comment from Franklin T. Adams-Watters, Aug 05 2012: Sketch of proof that the above sum of primes results in this sequence. If S_n is the set of possible values of the signed sums for the first n primes, then S_{n+1} = S_n U (S_n + prime(n+1)) U (S_n - prime(n+1)). Beyond about n=4, this will be everything even or everything odd in an interval around zero, and then a fringe on either side; the size of the interval will be 2 * A007504(n) - k for some small k. Recursively, since prime(n) << A007504(n), this will continue to hold. Hence the sequence continues to alternate 0's and 1's.  A quite modest estimate on the distribution of primes suffices to complete the proof.
For number of solutions see A022894, A113040; also A083309.

StackExchange, A prime number pattern, Jul 29 2012.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A007504, A214912, A215029, A215030; A022894, A083309, A113040.

Essentially the same as A135528, A059841, A000035.

PadRight[{2},120,{0,1}] (* Harvey P. Dale, Nov 29 2018 *)

if(n%2,2*(n<2),1) \\ Charles R Greathouse IV, Aug 06 2012

A215221 Number of solutions to p(n) = Sum_{i=1..n-1} c(i)*p(i) with c(i) in {-1,0,1} and p(n) = n-th prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 5, 11, 28, 69, 164, 437, 1104, 2887, 7778, 20861, 55610, 148857, 408694, 1112103, 3059571, 8519916, 23586160, 65766961, 183122954, 508287720, 1423807763, 4019399991, 11359914488, 32294035715, 91866217942, 258134484981, 732226048291

Offset: 1

Alois P. Heinz, Aug 06 2012

nonn

			a(3) = 1: prime(3) = 5 = 3+2.
a(4) = 1: prime(4) = 7 = 5+2.
a(5) = 1: prime(5) = 11 = 7+5-3+2.
a(6) = 5: prime(6) = 13 = 7+5+3-2 = 11+2 = 11+5-3 = 11+7-3-2 = 11+7-5.
a(7) = 11: prime(7) = 17 = 7+5+3+2 = 11+5+3-2 = 11+7-3+2 = 13+5-3+2 = 13+7-3 = 13+7-5+2 = 13-11+7+5+3 = 13+11-5-2 = 13+11-7 = 13+11-7-5+3+2 = 13+11-7+5-3-2.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..1000 (first 200 terms from Alois P. Heinz)

Cf. A000040, A007504, A022894, A083309, A113040, A215222.

sp:= proc(n) option remember; `if`(n=0, 0, ithprime(n)+sp(n-1)) end:
b := proc(n, i) option remember; `if`(n>sp(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(ithprime(n), n-1):
seq(a(n), n=1..40);

nmax = 40; d = {1}; a1 = {};
Do[
  p = Prime[n];
  i = Ceiling[Length[d]/2] + p;
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 p] + PadRight[d, Length[d] + 2 p] +
    PadLeft[PadRight[d, Length[d] + p], Length[d] + 2 p];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 11 2014 *)

a(n) = A215222(A000040(n)).

A215222 Number of solutions to n = Sum_{i=1..pi(n-1)} c(i)*p(i) with c(i) in {-1,0,1}, p(n) = n-th prime and pi = A000720.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 1, 5, 5, 13, 12, 11, 11, 29, 28, 74, 73, 71, 69, 184, 182, 176, 173, 170, 164, 446, 437, 1180, 1165, 1147, 1137, 1115, 1104, 2984, 2949, 2919, 2887, 7841, 7778, 21331, 21184, 21029, 20861, 57465, 57114, 56741, 56372, 55997, 55610

Offset: 1

Alois P. Heinz, Aug 06 2012

nonn

			a(5) = 1: 5 = 3+2.
a(6) = 1: 6 = 5+3-2.
a(7) = 1: 7 = 5+2.
a(8) = 2: 8 = 5+3 = 7+3-2.
a(9) = 2: 9 = 7+2 = 7+5-3.
a(10) = 3: 10 = 5+3+2 = 7+3 = 7+5-2.
a(11) = 1: 11 = 7+5-3+2.
a(12) = 5: 12 = 7+3+2 = 7+5 = 11+3-2 = 11-7+5+3 = 11+7-5-3+2.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A007504, A022894, A083309, A113040, A215221.

sp:= proc(n) option remember; `if`(n=0, 0, ithprime(n)+sp(n-1)) end:
b := proc(n, i) option remember; `if`(n>sp(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, numtheory[pi](n-1)):
seq(a(n), n=1..60);

sp[n_] := sp[n] = If[n == 0, 0, Prime[n]+sp[n-1]]; b[n_, i_] := b[n, i] = If[n>sp[i], 0, If[i == 0, 1, b[n, i-1] + b[n+Prime[i], i-1] + b[Abs[n-Prime[i]], i-1]]]; a[n_] := b[n, PrimePi[n-1]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 03 2014, after Alois P. Heinz *)

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

Original entry on oeis.org

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)).

A350881 a(n) is the constant term in expansion of Product_{k=1..n} (x^prime(k) + 1/x^prime(k))^2.

Original entry on oeis.org

1, 2, 4, 10, 24, 50, 140, 368, 1152, 3682, 11784, 39902, 134612, 463066, 1635092, 5818384, 20684072, 73693068, 266943648, 967762792, 3533666568, 13036452946, 48102671884, 178315730764, 661567489568, 2450447537226, 9123572154720, 34201574126260

Offset: 0

Ilya Gutkovskiy, Jan 20 2022

nonn

Cf. A000980, A022894, A022896, A047653.

p:= proc(n) option remember; `if`(n=0, 1,
      p(n-1)*(x^ithprime(n)+1/x^ithprime(n))^2)
    end:
a:= n-> coeff(p(n), x, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 26 2022

p[n_] := p[n] = If[n == 0, 1, p[n - 1]*(x^Prime[n] + 1/x^Prime[n])^2];
a[n_] := Coefficient[p[n], x, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

a(n) = polcoef (prod(k=1, n, (x^prime(k) + 1/x^prime(k))^2), 0); \\ Michel Marcus, Jan 21 2022

cp's OEIS Frontend

A350404 Number of solutions to +-2 +- 3 +- 5 +- 7 +- ... +- prime(n) = 0 or 1.

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A022900 Number of solutions to c(1)*prime(3) + ... + c(n)*prime(n+2) = 0, where c(i) = +-1 for i>1, c(1) = 1.

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A261062 Number of solutions to c(1)*prime(2) + ... + c(2n-1)*prime(2n) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

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A292698 Number of solutions to 3 +- 7 +- 13 +- 19 +- ... +- prime(4*n) = 0.

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A292699 Number of solutions to 5 +- 11 +- 17 +- 23 +- ... +- prime(4*n+1) = 0.

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A215036 2 followed by "1,0" repeated.

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A215221 Number of solutions to p(n) = Sum_{i=1..n-1} c(i)*p(i) with c(i) in {-1,0,1} and p(n) = n-th prime.

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A022900 Number of solutions to c(1)prime(3) + ... + c(n)prime(n+2) = 0, where c(i) = +-1 for i>1, c(1) = 1.

A261062 Number of solutions to c(1)prime(2) + ... + c(2n-1)prime(2n) = -1, where c(i) = +-1 for i > 1, c(1) = 1.