A113040 -id:A113040 - cp's OEIS frontend

A022894 Number of solutions to c(1)prime(1) +...+ c(2n+1)prime(2n+1) = 0, where c(i) = +-1 for i > 1, c(1) = 1.

0, 1, 1, 2, 5, 13, 39, 122, 392, 1286, 4341, 14860, 51085, 178402, 634511, 2260918, 8067237, 29031202, 105250449, 383579285, 1404666447, 5171065198, 19141008044, 71124987313, 263548339462, 983424096451, 3684422350470, 13818161525284, 51938115653565

Offset: 0

Clark Kimberling

c(1)*prime(1) + ... + c(2n)*prime(2n) = 0 has no solution, because the l.h.s. has an odd number of odd terms and the r.h.s. is even.

			a(1) = 1 because 2 + 3 - 5 = 0,
a(2) = 1 because 2 - 3 + 5 + 7 - 11 = 0,
a(3) = 2 because
  2 + 3 - 5 - 7 + 11 + 13 - 17 =
  2 + 3 - 5 + 7 - 11 - 13 + 17 = 0.
a(4) = 5 because
  2 - 3 - 5 + 7 + 11 + 13 + 17 - 19 - 23 =
  2 - 3 + 5 - 7 + 11 + 13 - 17 + 19 - 23 =
  2 - 3 + 5 + 7 - 11 - 13 + 17 + 19 - 23 =
  2 - 3 + 5 + 7 - 11 + 13 - 17 - 19 + 23 =
  2 + 3 + 5 - 7 - 11 - 13 + 17 - 19 + 23 = 0
and there are no others up through the ninth prime.

Ray Chandler, Table of n, a(n) for n = 0..1000 (first 101 terms from T. D. Noe)

Cf. A113040, A215036, A083309 (sums of odd primes).
Cf. A022895, A022896 (r.h.s. = 1 & 2, using all primes), A083309 and A022897 - A022899 (using primes >= 3), A022900 - A022902 (using primes >=5), A022903, A022904, A022920 (using primes >= 7); A261061 - A261063 & A261045 (r.h.s. = -1); A261057, A261059, A261060 & A261044 (r.h.s. = -2).
Bisection (odd part) of A306443.

sp:= proc(n) sp(n):= `if`(n=1, 0, ithprime(n)+sp(n-1)) end:
b := proc(n,i) option remember; `if`(n>sp(i), 0, `if`(i=1, 1,
        b(n+ithprime(i), i-1)+ b(abs(n-ithprime(i)), i-1)))
     end:
a:= n-> b(2, 2*n+1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 05 2012

Do[a = Table[ Prime[i], {i, 1, n} ]; c = 0; k = 2^(n - 1); While[k < 2^n, If[ Apply[ Plus, a*(-1)^(IntegerDigits[k, 2] + 1)] == 0, c++ ]; k++ ]; Print[c], {n, 1, 32, 2} ]

A022894={a(n, s=0-prime(1), p=1)=if(n<=s, if(s==p, n==s, a(abs(n-p), s-p, precprime(p-1))+a(n+p, s-p, precprime(p-1))), if(s<=0, a(abs(s), max(sum(i=p+1, p+(p>1)+2*n, prime(i)),1), prime(p+(p>1)+2*n))))} \\ M. F. Hasler, Aug 09 2015

Conjecture: limit_{n->oo} a(n)^(1/n) = 4. - Vaclav Kotesovec, Jun 05 2019

a(n) is the constant term in expansion of (1/2) * Product_{k=1..2*n+1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 25 2024

Edited by Robert G. Wilson v, Jan 29 2002
More terms from T. D. Noe, Jan 16 2007
Edited by M. F. Hasler, Aug 09 2015

A261057 Number of solutions to c(1)prime(1)+...+c(2n-1)prime(2n-1) = -2, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

0, 0, 1, 1, 5, 13, 40, 123, 388, 1284, 4332, 14868, 51094, 178361, 634422, 2260717, 8066841, 29030051, 105247340, 383574146, 1404657053, 5171018981, 19140750300, 71124341227, 263546155710, 983417309702, 3684399940711, 13818092760075, 51937827473594, 195956606402526

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 there would be an odd number of odd terms but the r.h.s. is even.

			a(1) = a(2) = 0 because prime(1) and prime(1) +- prime(2) +- prime(3) is always different from -2.
a(3) = 1 because prime(1) - prime(2) - prime(3) - prime(4) + prime(5) = -2.

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

Cf. A261059, A261060, A261045 (starting with prime(2) - prime(4)), A261061 - A261063 and A261044 (r.h.s. = -1), A022894 - A022904, A083309, A022920 (r.h.s. = 0, 1 or 2); A113040, A113041, A113042.

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

s[n_] := s[n] = If[n<2, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 1, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]]; a[n_] := b[4, 2*n-1];  Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

A261057(n,rhs=-2,firstprime=1)={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, s=-2-prime(1), p=1)={if(n<=s, if(s==p, n==s, a(abs(n-p), s-p, precprime(p-1))+a(n+p, s-p, precprime(p-1))), if(s<=0, a(abs(s), max(sum(i=p+1, p+=2*n-2+bittest(s,0), prime(i)),1), prime(p))))} \\ M. F. Hasler, Aug 09 2015

a(n) = A113041(n) - A022896(2n-1).

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

a(26)-a(30) from Alois P. Heinz, Jan 04 2019

A022895 Number of solutions to c(1)prime(1) + ... + c(n)prime(n) = 1, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 0, 8, 0, 22, 0, 70, 0, 218, 0, 708, 0, 2354, 0, 8015, 0, 27561, 0, 95160, 0, 335579, 0, 1202236, 0, 4267477, 0, 15318171, 0, 55248419, 0, 200711050, 0, 733704990, 0, 2696599982, 0, 9941660942, 0, 36928370497, 0, 136801720627, 0

Offset: 1

Clark Kimberling

nonn

			a(8) counts these 3 solutions: {2, -3, -5, 7, -11, 13, 17, -19}, {2, -3, -5, 7, 11, -13, -17, 19}, {2, -3, 5, -7, -11, 13, -17, 19}.

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

Cf. A022894 (r.h.s. = 0), A022896, ..., A022904, A083309, A022920 (variants with r.h.s. in {0, 1 or 2}, starting with prime(2) or prime(3) or prime(4)).

Cf. A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060 and A261044 (r.h.s. = -2); A113040 - A113042.

{f, s} = {1, 1}; 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}]
(* A022895, 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 3 solutions using n=8 primes; Peter J. C. Moses, Oct 01 2013 *)

A022895(n, rhs=1, firstprime=1)={rhs-=prime(firstprime); my(p=vector(n-1, 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 >> 20. - M. F. Hasler, Aug 08 2015

a(n, s=1-prime(1), p=1)={if(n<=s, if(s==p, n==s, a(abs(n-p), s-p, precprime(p-1))+a(n+p, s-p, precprime(p-1))), if(s<=0, if(n>1,a(abs(s), sum(i=p+1, p+n-1, prime(i)), prime(p+n-1)),!s)))} \\ On function call, s = r.h.s.- smallest prime; during recursion: sum of all primes to be used. - M. F. Hasler, Aug 09 2015

a(n) = [x^1] Product_{k=2..n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 26 2024

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

A022896 Number of solutions to c(1)prime(1) + ... + c(n)prime(n) = 2, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 0, 4, 0, 14, 0, 38, 0, 126, 0, 394, 0, 1290, 0, 4344, 0, 14846, 0, 51068, 0, 178436, 0, 634568, 0, 2261052, 0, 8067296, 0, 29031484, 0, 105251904, 0, 383580180, 0, 1404666680, 0, 5171079172, 0, 19141098744, 0, 71125205900, 0, 263549059326

Offset: 1

Clark Kimberling

nonn

			a(7) counts these 2 solutions: {2, -3, -5, -7, 11, -13, 17}, {2, 3, 5, 7, -11, 13, -17}.

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

Cf. A022894 (r.h.s. = 0), A022895 (r.h.s. = 1), A022897, ..., A022904, A022920 (using primes >= 7), A083309; A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060 and A261044 (r.h.s. = -2); A113040, A113041, A113042. - M. F. Hasler, Aug 08 2015

{f, s} = {1, 2}; 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}]
(* A022896, a(n) = number of solutions of "sum = s" using Prime(f) to Prime(f+n-1) *)
n = 7; 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 2 solutions of using n=7 primes; Peter J. C. Moses, Oct 01 2013 *)

A022896(n, rhs=2, firstprime=1)={rhs-=prime(firstprime); my(p=vector(n-1, i, prime(i+firstprime))); !(rhs||#p)+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 >> 20. - M. F. Hasler, Aug 08 2015

a(n,s=2-prime(1),p=1)={if(n<=s,if(s==p,n==s,a(abs(n-p),s-p,precprime(p-1))+a(n+p,s-p,precprime(p-1))),if(s<=0,if(n>1,a(abs(s),sum(i=p+1,p+n-1,prime(i)),prime(p+n-1)),!s)))} \\ M. F. Hasler, Aug 09 2015

a(2n-1) = A113041(n) - A261057(n), a(2n) = 0 because there is an odd number of odd terms on the left hand side, but the right hand side is even. - M. F. Hasler, Aug 09 2015

a(n) = [x^0] Product_{k=2..n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 26 2024

Corrected and extended by Clark Kimberling, Oct 01 2013

a(23)-a(49) from Alois P. Heinz, Aug 06 2015

A306443 Number of ways of partitioning the set of the first n primes into two subsets whose sums differ at most by 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 2, 6, 5, 16, 13, 45, 39, 138, 122, 439, 392, 1417, 1286, 4698, 4341, 16021, 14860, 55146, 51085, 190274, 178402, 671224, 634511, 2404289, 2260918, 8535117, 8067237, 30635869, 29031202, 110496946, 105250449, 401422210, 383579285, 1467402238

Offset: 0

Alois P. Heinz, May 31 2019

nonn

			a(8) = 6: 2,17,19/3,5,7,11,13; 3,5,11,19/2,7,13,17; 3,5,13,17/2,7,11,19; 3,7,11,17/2,5,13,19; 2,3,5,11,17/7,13,19; 2,5,7,11,13/3,17,19.
a(9) = 5: 2,3,5,17,23/7,11,13,19; 2,5,7,13,23/3,11,17,19; 2,5,7,17,19/3,11,13,23; 2,5,11,13,19/3,7,17,23; 2,7,11,13,17/3,5,19,23.

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

Bisections give: A022894 (odd part), A113040 (even part).

Cf. A000040, A069918, A307877.

s:= proc(n) s(n):= `if`(n=0, 1, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(i=0, `if`(n<=1, 1, 0),
     `if`(n>s(i), 0, (p->b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(i))))
    end:
a:= n-> ceil(b(0, n)/2):
seq(a(n), n=0..45);

s[n_] := s[n] = If[n == 0, 1, Prime[n] + s[n - 1]];
b[n_, i_] := b[n, i] = If[i==0, If[n <= 1, 1, 0], If[n > s[i], 0, Function[ p, b[n + p, i - 1] + b[Abs[n - p], i - 1]][Prime[i]]]];
a[n_] := Ceiling[b[0, n]/2];
a /@ Range[0, 45] (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

A113042 Number of solutions to +-p(1)+-p(2)+-...+-p(2n) = 3 where p(i) is the i-th prime.

Original entry on oeis.org

0, 2, 1, 7, 15, 45, 139, 438, 1419, 4703, 16019, 55146, 190254, 671215, 2404179, 8534995, 30635448, 110495549, 401418693, 1467388464, 5393131894, 19883104535, 73856058401, 273600682457, 1017557492609, 3803885439979, 14266466901249, 53564801078049

Offset: 1

Floor van Lamoen, Oct 12 2005

nonn

+-p(1)+-p(2)+-...+-p(2n+1) = 3 does not have solutions, since the left hand side is even. [Corrected and edited by M. F. Hasler, Aug 09 2015]

Ray Chandler, Table of n, a(n) for n = 1..1000 (first 120 terms from Alois P. Heinz)

Cf. A022894 - A022904, A022920, A083309; A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060 and A261044 (r.h.s. = -2); A113040, A113041.

A113042:=proc(n) local i,j,p,t; t:= NULL; for j from 2 to 2*n by 2 do p:=1; for i to j do p:=p*(x^(-ithprime(i))+x^(ithprime(i))); od; t:=t,coeff(p,x,3); od; t; end;
# second Maple program
sp:= proc(n) sp(n):= `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+ithprime(i), i-1)+ b(abs(n-ithprime(i)), i-1)))
     end:
a:= n-> b(3, 2*n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 05 2012

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 + Prime[i], i-1] + b[Abs[n - Prime[i]], i-1]]]; a[n_] := b[3, 2*n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 31 2017, after Alois P. Heinz *)

a(n) = [x^3] Product_{k=1..2*n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 30 2024

A113041 Number of solutions to +-p(1)+-p(2)+-...+-p(2n-1) = 2, where p(i) is the i-th prime.

Original entry on oeis.org

1, 0, 1, 3, 9, 27, 78, 249, 782, 2574, 8676, 29714, 102162, 356797, 1268990, 4521769, 16134137, 58061535, 210499244, 767154326, 2809323733, 10342098153, 38281849044, 142249547127, 527095215036, 1966843667482, 7368829743507, 27636276043171, 103876045792060

Offset: 1

Floor van Lamoen, Oct 12 2005

nonn

+-p(1)+-p(2)+-...+-p(2n) = 2 has no solutions, since the left hand side is odd.

Ray Chandler, Table of n, a(n) for n = 1..1000 (first 120 terms from Alois P. Heinz)

Cf. A022894 - A022904, A022920, A083309; A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060 and A261044 (r.h.s. = -2); A113040, A113042.

A113041:=proc(n) local i,j,p,t; t:= NULL; for j to 2*n-1 by 2 do p:=1; for i to j do p:=p*(x^(-ithprime(i))+x^(ithprime(i))); od; t:=t,coeff(p,x,2); od; t; end;
# second Maple program
sp:= proc(n) sp(n):= `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+ithprime(i), i-1)+ b(abs(n-ithprime(i)), i-1)))
     end:
a:= n-> b(2, 2*n-1):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 05 2012

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 + Prime[i], i-1] + b[Abs[n - Prime[i]], i-1]]];
a[n_] := b[2, 2n-1];
Array[a, 30] (* Jean-François Alcover, Nov 02 2020, after Alois P. Heinz *)

a(n) = A022896(2n-1) + A261057(n). - M. F. Hasler, Aug 09 2015

a(n) = [x^2] Product_{k=1..2*n-1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 30 2024

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

Original entry on oeis.org

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

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)).

cp's OEIS Frontend

A022894 Number of solutions to c(1)*prime(1) +...+ c(2n+1)*prime(2n+1) = 0, where c(i) = +-1 for i > 1, c(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A261057 Number of solutions to c(1)*prime(1)+...+c(2n-1)*prime(2n-1) = -2, where c(i) = +-1 for i > 1, c(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A022895 Number of solutions to c(1)*prime(1) + ... + c(n)*prime(n) = 1, where c(i) = +-1 for i > 1, c(1) = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A022896 Number of solutions to c(1)*prime(1) + ... + c(n)*prime(n) = 2, where c(i) = +-1 for i > 1, c(1) = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A306443 Number of ways of partitioning the set of the first n primes into two subsets whose sums differ at most by 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A113042 Number of solutions to +-p(1)+-p(2)+-...+-p(2n) = 3 where p(i) is the i-th prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A113041 Number of solutions to +-p(1)+-p(2)+-...+-p(2n-1) = 2, where p(i) is the i-th prime.

Views

Author

A022894 Number of solutions to c(1)prime(1) +...+ c(2n+1)prime(2n+1) = 0, where c(i) = +-1 for i > 1, c(1) = 1.

A261057 Number of solutions to c(1)prime(1)+...+c(2n-1)prime(2n-1) = -2, where c(i) = +-1 for i > 1, c(1) = 1.

A022895 Number of solutions to c(1)prime(1) + ... + c(n)prime(n) = 1, where c(i) = +-1 for i > 1, c(1) = 1.

A022896 Number of solutions to c(1)prime(1) + ... + c(n)prime(n) = 2, where c(i) = +-1 for i > 1, c(1) = 1.