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.

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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

A022903 Number of solutions to c(1)*prime(4) + ... + c(n)*prime(n+3) = 0, where c(i) = +-1 for i>1, c(1) = 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 6, 0, 9, 0, 61, 0, 131, 0, 486, 0, 2029, 0, 5890, 0, 21127, 0, 75979, 0, 273657, 0, 1032161, 0, 3694665, 0, 12989200, 0, 48409376, 0, 174262116, 0, 642786775, 0, 2402713235, 0, 8918299277, 0, 32868170524, 0, 123143998606, 0
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			a(10) counts these 6 solutions: {7, -11, -13, -17, -19, -23, 29, -31, 37, 41}, {7, 11, -13, 17, 19, -23, 29, 31, -37, -41}, {7, 11, -13, 17, 19, 23, -29, -31, 37, -41}, {7, 11, 13, -17, -19, 23, 29, 31, -37, -41}, {7, 11, 13, -17, 19, 23, -29, -31, -37, 41}, {7, 11, 13, 17, -19, -23, 29, -31, 37, -41}.

Links

Crossrefs

Cf. A022894, A022895, ..., A022904, A083309, A022920 (variants with r.h.s. in {0, 1 or 2}, starting with prime(1), prime(2), prime(3) or prime(4)); A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060, A261045(r.h.s. = -2).

Programs

  • Maple
    A022903 := 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) = 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
for n from 1 do
    print(n,A022903(n)) ;
end do: # R. J. Mathar, Aug 06 2015
  • Mathematica
    {f, s} = {4, 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}]
(* A022903, a(n) = number of solutions of "sum = s" using Prime(f) to Prime(f+n-1) *)
n = 10; 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 6 solutions of using n=10 primes; Peter J. C. Moses, Oct 01 2013 *)
  • PARI
    A022903(n, rhs=0, firstprime=4)={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

Formula

a(2n-1) = 0 for all n >= 1 because an odd number of odd terms on the l.h.s. cannot sum to zero. - M. F. Hasler, Aug 08 2015
a(n) = [x^7] Product_{k=5..n+3} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 28 2024

Extensions

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

A022897 Number of solutions to c(1)*prime(2) +...+ c(n)*prime(n+1) = 0, where c(i) = +-1 for i > 1, c(1) = 1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

Examples

			a(8) counts these 2 solutions: {3, 5, -7, 11, 13, 17, -19, -23}, {3, 5, 7, 11, -13, -17, -19, 23}. - _Clark Kimberling_, Oct 01 2013

Links

Crossrefs

Cf. A083309 (without odd n).
Cf. A022894 (use all primes in the sum), A022895 (r.h.s. = 1), A022896 (r.h.s. = 2),..., A022903 (using primes >= 7), A022904, A022920; A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060, A261044 (r.h.s. = -2).

Programs

  • Mathematica
    Table[ps = Prime[Range[2, n+1]]; pr = Inner[Times, 2 IntegerDigits[Range[2^(n-1), 2^n - 1], 2, n] - 1, ps, Plus]; Count[pr, 0], {n, 16}] (* T. D. Noe, Sep 30 2013 *)
  • PARI
    padbin(n, len) = {if (n, b = binary(n), b = [0]); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {nbs = 0; for (i = 2^(n-1), 2^n-1, vec = padbin(i, n); if (sum(k=1, n, if (vec[k], prime(k+1), -prime(k+1))) == 0, nbs++);); nbs;} \\ Michel Marcus, Sep 30 2013
  • PARI
    A022897(n, rhs=0, firstprime=2)={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
  • PARI
    a(n, s=0-3, 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+n-1, prime(i)), prime(p+n-1)))) \\ M. F. Hasler, Aug 09 2015

Formula

a(2n-1) = 0 (odd number of odd terms on the l.h.s.); a(2n) = A083309(n). - M. F. Hasler, Aug 08 2015
a(n) = [x^3] Product_{k=3..n+1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 26 2024

Extensions

a(20)-a(24) from Michel Marcus, Sep 30 2013
More terms from T. D. Noe, Sep 30 2013

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

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

Views

Author

Floor van Lamoen, Oct 12 2005

Keywords

Comments

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

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, A113042.

Programs

  • Maple
    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
  • 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[2, 2n-1];
Array[a, 30] (* Jean-François Alcover, Nov 02 2020, after Alois P. Heinz *)

Formula

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

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

Views

Author

Clark Kimberling

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    {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 *)
  • PARI
    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

Formula

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

Extensions

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

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 all terms are odd but the r.h.s. is odd, too.

Examples

			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).

Links

Crossrefs

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).

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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.

Formula

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

Extensions

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