cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

M. F. Hasler, Aug 08 2015

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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.
  • PARI
    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

Formula

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

Extensions

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

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

Views

Author

Clark Kimberling

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    {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 *)
  • PARI
    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
  • PARI
    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

Formula

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

Extensions

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

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

Views

Author

Floor van Lamoen, Oct 12 2005

Keywords

Comments

+-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]

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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

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