A022895 -id:A022895 - 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

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

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

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

Clark Kimberling

nonn

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

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

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

{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 *)

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

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

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

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

Clark Kimberling

nonn

			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}.

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

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).

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

{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 *)

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

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

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

Clark Kimberling

nonn

			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

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

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).

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 *)

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

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

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

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

a(20)-a(24) from Michel Marcus, Sep 30 2013

More terms from T. D. Noe, Sep 30 2013

A367088 Number of solutions to +- 1 +- 2 +- 3 +- 5 +- 7 +- ... +- prime(n-1) = 0 or 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 6, 7, 12, 19, 32, 53, 90, 156, 276, 493, 878, 1566, 2834, 5146, 9396, 17358, 32042, 59434, 110292, 204332, 380548, 713601, 1342448, 2538012, 4808578, 9043605, 17070234, 32268611, 61271738, 116123939, 220993892, 421000142, 802844420, 1534312896

Offset: 0

Ilya Gutkovskiy, Jan 26 2024

nonn

Cf. A008578, A025591, A022894, A022895, A113040, A350404.

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

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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.

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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.

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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.

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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.

A022896 Number of solutions to c(1)prime(1) + ... + c(n)prime(n) = 2, where c(i) = +-1 for i > 1, c(1) = 1.

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.

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.