A083309 -id:A083309 - cp's OEIS frontend

A292698 Number of solutions to 3 +- 7 +- 13 +- 19 +- ... +- prime(4*n) = 0.

0, 0, 0, 1, 2, 8, 27, 83, 292, 944, 3279, 11291, 38992, 138066, 490248, 1757360, 6347321, 22998089, 83780199, 306819363, 1128999790, 4174251748, 15507225620, 57767819903, 215327188611, 803901214851, 3013081897103, 11331883386143, 42737620941612

Offset: 1

Seiichi Manyama, Sep 21 2017

nonn

			For n = 4 the solution is 3 - 7 - 13 + 19 - 29 + 37 + 43 - 53 = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..200

Cf. A000040, A022894, A083309, A292699, A292700.

s:= proc(n) s(n):= `if`(n=0, 0, ithprime(2*n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      (p-> b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(2*i))))
    end:
a:= n-> b(0, 2*n)/2:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 21 2017

s[n_] := s[n] = If[n == 0, 0, Prime[2n] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1,
     With[{p = Prime[2i]}, b[n+p, i-1] + b[Abs[n-p], i-1]]]];
a[n_] := b[0, 2n]/2;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)

{a(n) = 1/2*polcoeff(prod(k=1, 2*n, x^prime(2*k)+1/x^prime(2*k)), 0)}

Constant term in the expansion of 1/2 * Product_{k=1..2*n} (x^prime(2*k) + 1/x^prime(2*k)).

A292699 Number of solutions to 5 +- 11 +- 17 +- 23 +- ... +- prime(4*n+1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 28, 102, 242, 835, 3381, 10115, 39415, 138555, 465778, 1741167, 6081563, 22156403, 81090107, 301185788, 1098440558, 4071519963, 15235221189, 56764430821, 211797564907, 790029575790, 2962705350767, 11154931819979, 42097079364709

Offset: 1

Seiichi Manyama, Sep 21 2017

nonn

			For n=2 the solution is 5-11-17+23 = 0.
For n=3 the solution is 5+11+17-23+31-41 = 0.
For n=4 the 2 solutions are 5-11+17+23+31+41-47-59 = 0 and 5+11-17+23+31-41+47-59 = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..200

Cf. A000040, A022894, A083309, A292698, A292700.

s:= proc(n) s(n):= `if`(n=0, 0, ithprime(2*n+1)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      (p-> b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(2*i+1))))
    end:
a:= n-> b(0, 2*n)/2:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 21 2017

s[n_] := s[n] = If[n == 0, 0, Prime[2n+1] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1,
     With[{p = Prime[2i+1]}, b[n+p, i-1] + b[Abs[n-p], i-1]]]];
a[n_] := b[0, 2n]/2;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)

{a(n) = 1/2*polcoeff(prod(k=1, 2*n, x^prime(2*k+1)+1/x^prime(2*k+1)), 0)}

Constant term in the expansion of 1/2 * Product_{k=1..2*n} (x^prime(2*k+1) + 1/x^prime(2*k+1)).

A215036 2 followed by "1,0" repeated.

Original entry on oeis.org

2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

N. J. A. Sloane, Aug 06 2012

Take the first n primes and combine them with coefficients +1 and -1; then a(n) is the smallest number (in absolute value) that can be obtained.
For example, a(1) = 2, a(2) = 1 from 3-2 = 1; a(3) = 0 from -2-3+5 = 0; a(11) = 0 from 2-3-5-7+11-13+17+19-23-29+31 = 0.
Comment from Franklin T. Adams-Watters, Aug 05 2012: Sketch of proof that the above sum of primes results in this sequence. If S_n is the set of possible values of the signed sums for the first n primes, then S_{n+1} = S_n U (S_n + prime(n+1)) U (S_n - prime(n+1)). Beyond about n=4, this will be everything even or everything odd in an interval around zero, and then a fringe on either side; the size of the interval will be 2 * A007504(n) - k for some small k. Recursively, since prime(n) << A007504(n), this will continue to hold. Hence the sequence continues to alternate 0's and 1's.  A quite modest estimate on the distribution of primes suffices to complete the proof.
For number of solutions see A022894, A113040; also A083309.

StackExchange, A prime number pattern, Jul 29 2012.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A007504, A214912, A215029, A215030; A022894, A083309, A113040.

Essentially the same as A135528, A059841, A000035.

PadRight[{2},120,{0,1}] (* Harvey P. Dale, Nov 29 2018 *)

if(n%2,2*(n<2),1) \\ Charles R Greathouse IV, Aug 06 2012

A215221 Number of solutions to p(n) = Sum_{i=1..n-1} c(i)*p(i) with c(i) in {-1,0,1} and p(n) = n-th prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 5, 11, 28, 69, 164, 437, 1104, 2887, 7778, 20861, 55610, 148857, 408694, 1112103, 3059571, 8519916, 23586160, 65766961, 183122954, 508287720, 1423807763, 4019399991, 11359914488, 32294035715, 91866217942, 258134484981, 732226048291

Offset: 1

Alois P. Heinz, Aug 06 2012

nonn

			a(3) = 1: prime(3) = 5 = 3+2.
a(4) = 1: prime(4) = 7 = 5+2.
a(5) = 1: prime(5) = 11 = 7+5-3+2.
a(6) = 5: prime(6) = 13 = 7+5+3-2 = 11+2 = 11+5-3 = 11+7-3-2 = 11+7-5.
a(7) = 11: prime(7) = 17 = 7+5+3+2 = 11+5+3-2 = 11+7-3+2 = 13+5-3+2 = 13+7-3 = 13+7-5+2 = 13-11+7+5+3 = 13+11-5-2 = 13+11-7 = 13+11-7-5+3+2 = 13+11-7+5-3-2.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..1000 (first 200 terms from Alois P. Heinz)

Cf. A000040, A007504, A022894, A083309, A113040, A215222.

sp:= proc(n) option remember; `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, i-1)+ b(n+ithprime(i), i-1)+ b(abs(n-ithprime(i)), i-1)))
     end:
a:= n-> b(ithprime(n), n-1):
seq(a(n), n=1..40);

nmax = 40; d = {1}; a1 = {};
Do[
  p = Prime[n];
  i = Ceiling[Length[d]/2] + p;
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 p] + PadRight[d, Length[d] + 2 p] +
    PadLeft[PadRight[d, Length[d] + p], Length[d] + 2 p];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 11 2014 *)

a(n) = A215222(A000040(n)).

A215222 Number of solutions to n = Sum_{i=1..pi(n-1)} c(i)*p(i) with c(i) in {-1,0,1}, p(n) = n-th prime and pi = A000720.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 1, 5, 5, 13, 12, 11, 11, 29, 28, 74, 73, 71, 69, 184, 182, 176, 173, 170, 164, 446, 437, 1180, 1165, 1147, 1137, 1115, 1104, 2984, 2949, 2919, 2887, 7841, 7778, 21331, 21184, 21029, 20861, 57465, 57114, 56741, 56372, 55997, 55610

Offset: 1

Alois P. Heinz, Aug 06 2012

nonn

			a(5) = 1: 5 = 3+2.
a(6) = 1: 6 = 5+3-2.
a(7) = 1: 7 = 5+2.
a(8) = 2: 8 = 5+3 = 7+3-2.
a(9) = 2: 9 = 7+2 = 7+5-3.
a(10) = 3: 10 = 5+3+2 = 7+3 = 7+5-2.
a(11) = 1: 11 = 7+5-3+2.
a(12) = 5: 12 = 7+3+2 = 7+5 = 11+3-2 = 11-7+5+3 = 11+7-5-3+2.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A007504, A022894, A083309, A113040, A215221.

sp:= proc(n) option remember; `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, i-1)+
        b(n+ithprime(i), i-1)+ b(abs(n-ithprime(i)), i-1)))
     end:
a:= n-> b(n, numtheory[pi](n-1)):
seq(a(n), n=1..60);

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, i-1] + b[n+Prime[i], i-1] + b[Abs[n-Prime[i]], i-1]]]; a[n_] := b[n, PrimePi[n-1]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 03 2014, after Alois P. Heinz *)

A238894 Irregular triangle of the number of times that sums +- 3 +- 5 +- 7 +- 11 +-...+- prime(2n+1) equal an even number in the range -d to d, where d = 3 + 5 + 7 + 11 +...+ prime(2n+1).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0

Offset: 1

T. D. Noe, Mar 07 2014

Because the value at odd numbers is zero, we count only the values at even numbers. This sequence, a generalization of A083309, is more interesting plotted.

The rows of the irregular triangle begin at positions 1, 10, 37, 94, 193, 352, 589, 916, 1355, 1922, 2633, 3506, 4565, 5828, and 7307. having lengths 9, 27, 57, 99, 159, 237, 327, 439, 567, 711, 873, 1059, 1263, 1479, and 1719.

			The first row of the irregular triangle is {1, 0, 0, 1, 0, 1, 0, 0, 1} because the sums +- 3 +- 5 form the numbers -8, -2, 2, and 8. The odd numbers are suppressed.

T. D. Noe, Rows n = 1..15 of irregular triangle, flattened
T. D. Noe, Extremal Sums of Sequences

Cf. A083309.

nMax = 10; d = {1, 0, 0, 1}; t = {}; Do[p = Prime[n + 1]; d = PadLeft[d, Length[d] + p] + PadRight[d, Length[d] + p]; If[0 == Mod[n, 2], AppendTo[t, d]], {n, 2, nMax}]; Flatten[t]

cp's OEIS Frontend

A292698 Number of solutions to 3 +- 7 +- 13 +- 19 +- ... +- prime(4*n) = 0.

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A292699 Number of solutions to 5 +- 11 +- 17 +- 23 +- ... +- prime(4*n+1) = 0.

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A215036 2 followed by "1,0" repeated.

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A215221 Number of solutions to p(n) = Sum_{i=1..n-1} c(i)*p(i) with c(i) in {-1,0,1} and p(n) = n-th prime.

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A215222 Number of solutions to n = Sum_{i=1..pi(n-1)} c(i)*p(i) with c(i) in {-1,0,1}, p(n) = n-th prime and pi = A000720.

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A238894 Irregular triangle of the number of times that sums +- 3 +- 5 +- 7 +- 11 +-...+- prime(2n+1) equal an even number in the range -d to d, where d = 3 + 5 + 7 + 11 +...+ prime(2n+1).

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