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