A022920 -id:A022920 - 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

A083309 a(n) is the number of times that sums 3 +- 5 +- 7 +- 11 +- ... +- prime(2n+1) of the first 2n odd primes is zero. There are 2^(2n-1) choices for the sign patterns.

Original entry on oeis.org

0, 0, 1, 2, 7, 19, 63, 197, 645, 2172, 7423, 25534, 89218, 317284, 1130526, 4033648, 14515742, 52625952, 191790090, 702333340, 2585539586, 9570549372, 35562602950, 131774529663, 491713178890, 1842214901398, 6909091641548

Offset: 1

T. D. Noe, Apr 29 2003

nonn

The frequency of each possible sum is computed by the Mathematica program without explicitly computing the individual sums. Let S = 3 + 5 + 7 + ... + prime(2n+1). Because the primes do not grow very fast, it is easy to show that, for n > 2, all even numbers between -S+20 and S-20 occur at least once as a sum.
a(n) is the maximal number of subsets of {prime(2), prime(3), ..., prime(n+1)} that share the same sum. Cf. A025591, A083527.
See A238894 for a more general sequence that looks at all sums formed. - T. D. Noe, Mar 07 2014

			a(3) = 1 because there is only one sign pattern of the first six odd primes that yields zero: 3 + 5 + 7 - 11 + 13 - 17.

Ray Chandler, Table of n, a(n) for n = 1..1000 (first 100 terms from T. D. Noe)
T. D. Noe, Extremal Sums of Sequences.

Cf. A015818, A063865, A238894.

Cf. A022894 (use all primes in the sum), A022895 (r.h.s. = 1), A022896 (r.h.s. = 2), A022897 (interleaved 0 for odd number of terms), ..., A022903 (using primes >= 7), A022904, A022920; A261061 - A261063 and A261044 (r.h.s. = -1); A261057, A261059, A261060, A261045 (r.h.s. = -2).

d={1, 0, 0, 1}; nMax=32; zeroLst={}; Do[p=Prime[n+1]; d=PadLeft[d, Length[d]+p]+PadRight[d, Length[d]+p]; If[0==Mod[n, 2], AppendTo[zeroLst, d[[(Length[d]+1)/2]]]], {n, 2, nMax}]; zeroLst/2

A083309(n, rhs=0, 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. - M. F. Hasler, Aug 08 2015

a(n) = A022897(2n). - M. F. Hasler, Aug 08 2015

A022904 Number of solutions to c(1)prime(4) + ... + c(n)prime(n+3) = 1, where c(i) = +-1 for i>1, c(1) = 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 6, 0, 32, 0, 110, 0, 252, 0, 1139, 0, 3127, 0, 12743, 0, 39767, 0, 156376, 0, 517381, 0, 1870169, 0, 6786580, 0, 25420402, 0, 90815872, 0, 334621081, 0, 1235976769, 0, 4597232973, 0, 17047065235, 0, 63450750049, 0, 238163814619, 0

Offset: 1

Clark Kimberling

nonn

			a(7) counts these 3 solutions: {7, -11, 13, 17, -19, 23, -29}, {7, 11, -13, -17, 19, 23, -29}, {7, 11, 13, -17, -19, -23, 29}.

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

Cf. A022903, A022920.

A022904 := proc(n)
    local a,b,cs,cslen ;
    a := 0 ;
    for b from 0 to 2^(n-1)-1 do
        cs := convert(b,base,2) ;
        cslen := nops(cs) ;
        if cslen < n-1 then
            cs := [op(cs),seq(0,i=1..n-1-cslen)] ;
        end if;
        if ithprime(4)+add( (-1+2*op(i-4,cs)) *ithprime(i),i=5..n+3) = 1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 06 2015

{f, s} = {4, 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}]
(* A022904, 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 3 solutions of using n=7 primes; Peter J. C. Moses, Oct 01 2013 *)

a(n) = [x^6] Product_{k=5..n+3} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 30 2024

Corrected and extended by Clark Kimberling, Oct 01 2013

a(23)-a(50) from Alois P. Heinz, Aug 06 2015

A261061 Number of solutions to c(1)prime(1)+...+c(2n)prime(2n) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

1, 0, 2, 3, 8, 23, 68, 221, 709, 2344, 8006, 27585, 95114, 335645, 1202053, 4267640, 15317698, 55248527, 200711160, 733697248, 2696576651, 9941588060, 36928160817, 136800727634, 508780005068, 1901946851732, 7133247301621, 26782446410398, 100862459737318

Offset: 1

M. F. Hasler, Aug 08 2015

nonn

There cannot be a solution for an odd number of terms on the l.h.s. because there would be an even number of odd terms but the r.h.s. is odd.

			a(1) = 1 counts the solution prime(1) - prime(2) = -1.
a(2) = 0 because prime(1) +- prime(2) +- prime(3) +- prime(4) is always different from -1.
a(3) = 2 counts the two solutions prime(1) - prime(2) + prime(3) - prime(4) - prime(5) + prime(6) = -1 and prime(1) - prime(2) - prime(3) + prime(4) + prime(5) - prime(6) = -1.

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

Cf. A261062 - A261063 and A261044 (starting with prime(2), prime(3) resp. prime(4)), A022894 - A022904, A083309, A022920 (r.h.s. = 0, 1 or 2), A261057, A261059, A261060, A261045 (r.h.s. = -2).

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(3, 2*n):
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[3, 2*n]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

A261061(n,rhs=-1,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.

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

a(n) = [x^3] Product_{k=2..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

A261045 Number of solutions to c(1)prime(4) + c(2)prime(5) + ... + c(2n-1)*prime(2n+2) = -1, where c(i) = +-1 for i>1, c(1) = 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 32, 93, 261, 1082, 3253, 12307, 40809, 153392, 525417, 1892876, 6847161, 25256461, 91268129, 335852960, 1239350769, 4606651034, 17073491494, 63523866957, 237953442636, 892247156886, 3346127378391, 12603121634857, 47642071407103

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 they are all odd and the r.h.s. is odd, too.

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

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

s:= proc(n) option remember;
      `if`(n<5, 0, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=4, 1,
      b(abs(n-ithprime(i)),i-1)+b(n+ithprime(i),i-1)))
    end:
a:= n-> b(8, 2*n+2):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 08 2015

s[n_] := s[n] = If[n<5, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 4, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]]; a[n_] := b[8, 2*n+2]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n)={my(p=vector(2*n-2,i,prime(i+4)));sum(i=1,2^(2*n-2),sum(j=1,#p,(1-bittest(i,j-1)<<1)*p[j],7)==-1)} \\ For illustrative purpose; too slow for n >> 10. - M. F. Hasler, Aug 08 2015

a(13)-a(29) from Alois P. Heinz, Aug 08 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

A261063 Number of solutions to c(1)prime(3) + ... + c(2n-1)prime(2n+1) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

0, 0, 0, 1, 6, 8, 40, 67, 373, 1232, 3330, 13656, 47111, 164957, 582042, 1967152, 7129046, 26655235, 94956602, 353789267, 1300061367, 4765080122, 17726643505, 66038899483, 245431428625, 919911458949, 3457983108462, 12974054097333, 49016641868213, 185510228030858

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(3) and prime(3) +- prime(4) +- prime(5) are different from -1 for any choice of the signs.
a(3) = 0 because the same sums prime(3) +- ... +- prime(7) is also always different from -1 for any choice of the signs.
a(4) = 1 because prime(3) - prime(4) - prime(5) - prime(6) - prime(7) + prime(8) + prime(9) = -1 is the only solution.

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

Cf. A261061 - A261062 (starting with prime(1) resp. prime(2)), A261044 (starting with prime(4)), A022894 - A022904, A083309, A022920 (r.h.s. = 0, 1 or 2), A261057, A261059, A261060, A261045 (r.h.s. = -2).

s:= proc(n) option remember;
      `if`(n<4, 0, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=3, 1,
      b(abs(n-ithprime(i)),i-1)+b(n+ithprime(i),i-1)))
    end:
a:= n-> b(6, 2*n+1):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 08 2015

s[n_] := s[n] = If[n<4, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 3, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]]; a[n_] := b[6, 2*n+1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

A261063(n,rhs=-1,firstprime=3)={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^6] Product_{k=4..2*n+1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024

a(15)-a(30) from Alois P. Heinz, Aug 08 2015

A261059 Number of solutions to c(1)prime(2)+...+c(2n)prime(2n+1) = -2, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

1, 0, 2, 1, 4, 25, 47, 237, 562, 1965, 7960, 24148, 85579, 307569, 1104519, 4106381, 14710760, 52113647, 193181449, 698356631, 2574590311, 9600573372, 35644252223, 131545038705, 492346772797, 1843993274342, 6903884199622, 25984680496124, 97937400336407

Offset: 1

M. F. Hasler, Aug 08 2015

nonn

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

			a(1) = 1 because prime(2) - prime(3) = -2.
a(2) = 0 because prime(2) +- prime(3) +- prime(4) +- prime(5) is different from -2 for any choice of the signs.
a(3) = 2 counts the 2 solutions prime(2) - prime(3) + prime(4) - prime(5) - prime(6) + prime(7) = -2 and prime(2) - prime(3) - prime(4) + prime(5) + prime(6) - prime(7) = -2.

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

Cf. A261057 (starting with prime(1)), A261060 (starting with prime(3)), A261045 (starting with prime(4)), A261061 - A261063 and A261044 (r.h.s. = -1), A022894 - A022904, A083309, A022920 (r.h.s. = 0, 1 or 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(5, 2*n+1):
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[5, 2*n+1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

A261059(n,rhs=-2,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,s=-2-3,p=2)=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),sum(i=p+1,p+2*n-1,prime(i)),prime(p+n*2-1))))

a(n) = [x^5] Product_{k=3..2*n+1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024

a(15)-a(29) from Alois P. Heinz, Aug 08 2015

A261060 Number of solutions to c(1)prime(3) + ... + c(2n)prime(2n+2) = -2, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

1, 0, 2, 1, 9, 22, 38, 143, 676, 1815, 7434, 22452, 87485, 290873, 1092072, 3894381, 13988849, 49672279, 184745525, 677809709, 2495632892, 9260315018, 34280441347, 127419049587, 474867366809, 1781565475308, 6700749901259, 25230023849115, 95215110677472

Offset: 1

M. F. Hasler, Aug 08 2015

nonn

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

			a(1) = 1 because prime(3) - prime(4) = -2.
a(2) = 0 because prime(3) +- prime(4) +- prime(5) +- prime(6) is different from -2 for any choice of the signs.
a(3) = 2 counts the two solutions prime(3) - prime(4) + prime(5) - prime(6) - prime(7) + prime(8) = 5 - 7 + 11 - 13 - 17 + 19 = -2 and prime(3) - prime(4) - prime(5) + prime(6) + prime(7) - prime(8) = 5 - 7 - 11 + 13 + 17 - 19 = -2.

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

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

s:= proc(n) option remember;
      `if`(n<4, 0, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=3, 1,
      b(abs(n-ithprime(i)),i-1)+b(n+ithprime(i),i-1)))
    end:
a:= n-> b(7, 2*n+2):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 08 2015

s[n_] := s[n] = If[n<4, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 3, 1, b[Abs[n-Prime[i]], i-1]+b[n+Prime[i], i-1]]];  a[n_] := b[7, 2*n+2]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n,rhs=-2,firstprime=3)={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^7] Product_{k=4..2*n+2} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024

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

A113040 Number of solutions to +-p(1)+-p(2)+-...+-p(2n)=1 where p(i) is the i-th prime.

Original entry on oeis.org

1, 1, 3, 6, 16, 45, 138, 439, 1417, 4698, 16021, 55146, 190274, 671224, 2404289, 8535117, 30635869, 110496946, 401422210, 1467402238, 5393176633, 19883249002, 73856531314, 273602448261, 1017563027699, 3803902663467, 14266523388813, 53564969402478

Offset: 1

Floor van Lamoen, Oct 12 2005

nonn

+-p(1)+-p(2)+-...+-p(2n+1)=1 has no solutions because the l.h.s. is even.

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

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..1000 (first 130 terms from Alois P. Heinz)

Cf. A083309, A022894 - A022904, A022920, A261057 - A261063 and A261044 - A261045; A215036.

Bisection (even part) of A306443.

A113040:=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,1); 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(1, 2*n):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 05 2012

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[1, 2*n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = A022895(2n) + A261061(n). - M. F. Hasler, Aug 09 2015
Conjecture: limit_{n->infinity} a(n)^(1/n) = 4. - Vaclav Kotesovec, Jun 05 2019
a(n) = [x^1] Product_{k=1..2*n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 25 2024

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.

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A083309 a(n) is the number of times that sums 3 +- 5 +- 7 +- 11 +- ... +- prime(2n+1) of the first 2n odd primes is zero. There are 2^(2n-1) choices for the sign patterns.

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A022904 Number of solutions to c(1)*prime(4) + ... + c(n)*prime(n+3) = 1, where c(i) = +-1 for i>1, c(1) = 1.

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A261061 Number of solutions to c(1)*prime(1)+...+c(2n)*prime(2n) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

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A261045 Number of solutions to c(1)*prime(4) + c(2)*prime(5) + ... + c(2n-1)*prime(2n+2) = -1, where c(i) = +-1 for i>1, c(1) = 1.

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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.

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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.

A022904 Number of solutions to c(1)prime(4) + ... + c(n)prime(n+3) = 1, where c(i) = +-1 for i>1, c(1) = 1.

A261061 Number of solutions to c(1)prime(1)+...+c(2n)prime(2n) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

A261045 Number of solutions to c(1)prime(4) + c(2)prime(5) + ... + c(2n-1)*prime(2n+2) = -1, 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.

A261063 Number of solutions to c(1)prime(3) + ... + c(2n-1)prime(2n+1) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

A261059 Number of solutions to c(1)prime(2)+...+c(2n)prime(2n+1) = -2, where c(i) = +-1 for i > 1, c(1) = 1.

A261060 Number of solutions to c(1)prime(3) + ... + c(2n)prime(2n+2) = -2, where c(i) = +-1 for i > 1, c(1) = 1.