A261044
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, 2, 0, 5, 0, 18, 0, 48, 0, 170, 0, 540, 0, 1868, 0, 6385, 0, 22247, 0, 79355, 0, 282754, 0, 1008714, 0, 3627599, 0, 13156851, 0, 47949883, 0, 175599692, 0, 646384942, 0, 2392644640, 0, 8890619925, 0, 32943781423, 0, 122928406923, 0
Offset: 1
a(8) = 2 counts the two solutions prime(4) - prime(5) + prime(6) - prime(7) - prime(8) + prime(9) - prime(10) + prime(11) = -2 and prime(4) - prime(5) - prime(6) + prime(7) + prime(8) - prime(9) - prime(10) + prime(11) = -2.
-
A261044(n, rhs=-2, 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 >> 10.
A083527
a(n) is the number of times that sums 1+-4+-9+-16+-...+-n^2 of the first n squares is zero. There are 2^(n-1) choices for the sign patterns.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 5, 0, 0, 43, 57, 0, 0, 239, 430, 0, 0, 2904, 5419, 0, 0, 27813, 50213, 0, 0, 348082, 649300, 0, 0, 3913496, 7287183, 0, 0, 50030553, 93696497, 0, 0, 611793542, 1161079907, 0, 0, 8009933135, 15176652567, 0, 0
Offset: 1
a(7) = 1 because there is only one sign pattern of the first seven squares that yields zero: 1+4-9+16-25-36+49.
-
b:= proc(n, i) option remember; local m;
m:= (1+(3+2*i)*i)*i/6;
`if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))
end:
a:= n-> `if`(irem(n-1, 4)<2, 0, b(n^2, n-1)):
seq(a(n), n=1..40); # Alois P. Heinz, Oct 31 2011
-
d={1, 1}; nMax=60; zeroLst={0}; Do[p=n^2; d=PadLeft[d, Length[d]+p]+PadRight[d, Length[d]+p]; If[1==Mod[Length[d], 2], AppendTo[zeroLst, d[[(Length[d]+1)/2]]], AppendTo[zeroLst, 0]], {n, 2, nMax}]; zeroLst/2
p = 1; t = {}; Do[p = Expand[p(x^(n^2) + x^(-n^2))]; AppendTo[t, Select[p, NumberQ[ # ] &]/2], {n, 51}]; t (* Robert G. Wilson v, Oct 31 2005 *)
-
a(n)=sum(i=0,2^(n-1)-1,sum(j=1,n-1,(-1)^bittest(i,j-1)*j^2)==n^2) \\ Charles R Greathouse IV, Nov 05 2012
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}.
-
{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 *)
-
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
-
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
-
{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
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
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}.
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
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
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
-
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
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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
A059871
Number of solutions to the equation p_i = (1+mod(i,2))*p_{i-1} +- p_{i-2} +- p_{i-3} +- ... +- 2 +- 1, where p_i is the i-th prime number (where p_1 = 2 and the "zeroth prime" p_0 is defined to be 1).
Original entry on oeis.org
1, 1, 1, 1, 1, 3, 3, 4, 6, 12, 16, 31, 46, 90, 140, 276, 449, 877, 1443, 2834, 4725, 9395, 16153, 32037, 55872, 110288, 190815, 380488, 672728, 1342395, 2434797, 4808180, 8579625, 17070112, 30858078, 61271317, 110926277, 220979544, 402354848
Offset: 1
For the first five primes we have only one solution for each: 2 = 2*1, 3 = 1*2 + 1*1, 5 = 2*3 - 1*2 + 1*1, 7 = 1*5 + 1*3 - 1*2 + 1*1, 11 = 2*7 - 1*5 + 1*3 - 1*2 + 1*1 and for the next prime 13, we have 3 solutions: 13 = 11-7+5+3+2-1 = 11+7-5-3+2+1 = 11+7-5+3-2-1.
- D. M. Burton, Elementary Number Theory.
- S. S. Pillai, "On some empirical theorem of Scherk", J. Indian Math. Soc. 17 (1927-28), pp. 164-171.
- W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
- Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..1000 (first 250 terms from Alois P. Heinz)
- J. L. Brown, Proof of Scherk's Conjecture on the Representation of Primes, Amer. Math. Monthly 74 (1967), 31-33.
- William Y. Lee, On the representation of integers, Math. Mag. 47 (1974), 150-152.
- H. F. Scherk, Bemerkungen über die Bildung der Primzahlen aus einander, Journal für die reine und angewandte Mathematik 10 (1883), pp. 201-208.
- H. F. Scherk, Bemerkungen über die Bildung der Primzahlen aus einander, Journal für die reine und angewandte Mathematik 10 (1883), pp. 201-208.
See
A059872 for the table of all solutions encoded as binary vectors and
A059873-
A059875 for specific sequences.
A059876 gives the function bin_prime_sum.
-
map(nops, primesums_primes_mult(16)); primesums_primes_mult := proc(upto_n) local a,b,i,n,p,t; a := []; for n from 1 to upto_n do b := []; p := ithprime(n); for i from (2^(n-1)) to ((2^n)-1) do t := bin_prime_sum(i); if(t = p) then b := [op(b),i]; fi; od; a := [op(a),b]; print(a); od; RETURN(a); end;
# second Maple program
p:= n-> `if`(n<0, 0, `if`(n=0, 1, ithprime(n))):
sp:= proc(n) sp(n):= `if`(n<0, 0, p(n)+sp(n-1)) end:
b := proc(n, i) option remember; `if`(n>sp(i), 0, `if`(i<0, 1,
b(n+p(i), i-1)+ b(abs(n-p(i)), i-1)))
end:
a:= n-> b(p(n) -(1+irem(n, 2))*p(n-1), n-2):
seq(a(n), n=1..40); # Alois P. Heinz, Aug 05 2012
-
nmax = 40; d = {1}; a1 = {}; pp = 1;
Do[
p = Prime[n];
i = Ceiling[Length[d]/2] + Abs[p - (1 + Mod[n, 2])*pp];
AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
d = PadLeft[d, Length[d] + 2 pp] + PadRight[d, Length[d] + 2 pp];
pp = p;
, {n, nmax}];
a1 (* Ray Chandler, Mar 11 2014 *)
More terms from Larry Reeves (larryr(AT)acm.org), Nov 20 2003
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
-
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
-
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 *)
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
-
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
-
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 *)
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
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).
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).
-
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
-
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 *)
-
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.
Comments