A083309 -id:A083309 - cp's OEIS frontend

A025591 Maximal coefficient of Product_{k<=n} (1 + x^k). Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 1.

1, 1, 1, 2, 2, 3, 5, 8, 14, 23, 40, 70, 124, 221, 397, 722, 1314, 2410, 4441, 8220, 15272, 28460, 53222, 99820, 187692, 353743, 668273, 1265204, 2399784, 4559828, 8679280, 16547220, 31592878, 60400688, 115633260, 221653776, 425363952, 817175698

Offset: 0

David W. Wilson

If k is allowed to approach infinity, this gives the partition numbers A000009.

a(n) is the maximal number of subsets of {1,2,...,n} that share the same sum.

T. D. Noe, Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000, first 201 terms from T. D. Noe, next 200 terms from Alois P. Heinz)
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance , Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.4.
Vlad-Florin Dragoi and Valeriu Beiu, Fast Reliability Ranking of Matchstick Minimal Networks, arXiv:1911.01153 [cs.DM], 2019.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
Erich Friedman and Mike Keith, Magic Carpets, J. Int Sequences, 3 (2000), Article 00.2.5.
Susumu Kubo, Partially Ordered Sets Corresponding to the Partition Problem, arXiv:2405.05544 [cs.DM], 2024. See pp. 2, 16.
Marco Mondelli, S. Hamed Hassani, and Rüdiger Urbanke, Construction of Polar Codes with Sublinear Complexity, arXiv preprint arXiv:1612.05295 [cs.IT], 2016-2017. See Sect. I.
Robert A. Proctor, Solution of two difficult combinatorial problems with linear algebra, American Mathematical Monthly 89, 721-734.
Blair D. Sullivan, On a conjecture of Adrica and Tomescu, J. Int. Sequences 16 (2013), Article 13.3.1.

Cf. A039828, A063865, A069918, A063866, A063867, A083309, A083527, A086376.

Cf. A053632, A160235, A359319, A359320.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(i=0, 1, b(n+i, i-1)+b(abs(n-i), i-1)))
    end:
a:=n-> b(0, n)+b(1, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 10 2014

f[n_, s_] := f[n, s]=Which[n==0, If[s==0, 1, 0], Abs[s]>(n*(n+1))/2, 0, True, f[n-1, s-n]+f[n-1, s+n]]; Table[Which[Mod[n, 4]==0||Mod[n, 4]==3, f[n, 0], Mod[n, 4]==1||Mod[n, 4]==2, f[n, 1]], {n, 0, 40}]
(* Second program: *)
p = 1; Flatten[{1, Table[p = Expand[p*(1 + x^n)]; Max[CoefficientList[p, x]], {n, 1, 50}]}] (* Vaclav Kotesovec, May 04 2018 *)
b[n_, i_] := b[n, i] = If[n > i(i+1)/2, 0, If[i == 0, 1, b[n+i, i-1] + b[Abs[n-i], i-1]]];
a[n_] := b[0, n] + b[1, n]; a /@ Range[0, 40] (* Jean-François Alcover, Feb 17 2020, after Alois P. Heinz *)

a(n)=if(n<0,0,polcoeff(prod(k=1,n,1+x^k),n*(n+1)\4))

from collections import Counter
def A025591(n):
    c = {0:1,1:1}
    for i in range(2,n+1):
        d = Counter(c)
        for k in c:
            d[k+i] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

a(n) = A063865(n) + A063866(n).
a(n) ~ sqrt(6/Pi) * 2^n / n^(3/2) [conjectured by Andrica and Tomescu (2002) and proved by Sullivan (2013)]. - Vaclav Kotesovec, Mar 17 2020
More precise asymptotics: a(n) ~ sqrt(6/Pi) * 2^n / n^(3/2) * (1 - 6/(5*n) + 589/(560*n^2) - 39/(50*n^3) + ...). - Vaclav Kotesovec, Dec 30 2022
a(n) = max_{k>=0} A053632(n,k). - Alois P. Heinz, Jan 20 2023

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.

Original entry on oeis.org

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

A022920 Number of solutions to c(1)prime(4) + ... + c(n)prime(n+3) = 2, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 7, 0, 12, 0, 61, 0, 131, 0, 472, 0, 2039, 0, 5924, 0, 21095, 0, 76058, 0, 274023, 0, 1032989, 0, 3694643, 0, 12987172, 0, 48417270, 0, 174274092, 0, 642785629, 0, 2402825962, 0, 8918414212, 0, 32868915523, 0, 123145191037, 0

Offset: 1

Clark Kimberling

nonn

Each second entry is 0 because the primes that are involved are all odd and the right hand side is even. - R. J. Mathar, Aug 06 2015

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

Cf. A022894, A022895, A022896 (r.h.s. = 0, 1 & 2, using all primes), A083309 and A022897 - A022899 (using primes >= 3), A022900 - A022902 (using primes >=5), A022903, A022904 (r.h.s. = 0 & 1, using primes >= 7); A261061 - A261063 & A261045 (r.h.s. = -1); A261057, A261059, A261060 & A261044 (r.h.s. = -2).

b[n_, s_, p_] := b[n, s, p] = If[n <= s, If[s == p, Boole[n == s], b[Abs[n - p], s - p, NextPrime[p - 1, -1]] + b[n + p, s - p, NextPrime[p - 1, -1] ]], If[s <= 0, b[Abs[s], Sum[Prime[i], {i, p + 1, p + n - 1}], Prime[p + n - 1]]]] /. Null -> 0; a[n_] := b[n, 2 - Prime[4], 4]; Array[a, 50] (* Jean-François Alcover, Feb 14 2018, after M. F. Hasler *)

A022920(n)={my(p=vector(n-1,i,prime(i+4)));sum(i=1,2^(n-1),sum(j=1,#p,(1-bittest(i,j-1)<<1)*p[j],7)==2)} \\ For illustrative purpose; too slow for n >> 20. - M. F. Hasler, Aug 08 2015

a(n, s=2-prime(4), p=4)=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+n-1, prime(i)), prime(p+n-1)))) \\ M. F. Hasler, Aug 09 2015

a(2n-1) = 0 for all n >= 1.

Corrected by R. J. Mathar, Aug 06 2015

a(22)-a(49) 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

A025591 Maximal coefficient of Product_{k<=n} (1 + x^k). Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 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.

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A022920 Number of solutions to c(1)*prime(4) + ... + c(n)*prime(n+3) = 2, 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.

A022920 Number of solutions to c(1)prime(4) + ... + c(n)prime(n+3) = 2, 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.