A008347 -id:A008347 - cp's OEIS frontend

A175117 Numbers k such that Sum_(i=1..k) prime(i)*(-1)^(i+1) is a square.

3, 9, 55, 181, 215, 459, 1361, 1417, 1623, 2047, 2445, 4685, 6687, 7353, 7785, 7925, 8489, 10333, 10515, 11179, 14087, 15227, 15829

Offset: 1

Ctibor O. Zizka, Feb 14 2010

Numbers k such that (-1)^k*A008347(k) is a perfect square. - R. J. Mathar, Feb 21 2010

			2 - 3 + 5 = 4, thus a(1)=3;
2 - 3 + 5 - 7 + 11 - 13 + 17 - 19 + 23 = 16, thus a(2)=9.

Harvey P. Dale, Table of n, a(n) for n = 1..300

Cf. A000040.

Flatten[Position[Accumulate[Times@@@Partition[Riffle[Prime[Range[ 20000]], {1,-1},{2,-1,2}],2]],?(IntegerQ[Sqrt[#]]&)]] (* _Harvey P. Dale, Mar 19 2013 *)

Six more terms from R. J. Mathar, Feb 21 2010

A213325 Number of ways to write n = q + sum_{k=1}^m(-1)^{m-k}p_k, where p_k is the k-th prime, and q is a practical number with q-4 and q+4 also practical.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 3, 1, 2, 4, 3, 2, 3, 4, 4, 3, 3, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 3, 5, 3, 2, 4, 6, 4, 3, 6, 7, 2, 2, 6, 6, 2, 2, 5, 7, 2, 2, 5, 6, 3, 3, 3, 7, 3, 2, 3, 7, 4, 5, 4, 8, 2, 5, 4, 6, 2, 4, 2, 5, 3, 5, 4

Offset: 1

Zhi-Wei Sun, Mar 03 2013

nonn

Conjecture: a(n)>0 for all n>8.

The author has verified this for n up to 5*10^6.

			a(11)=1 since 11=8+(7-5+3-2) with 4, 8, 12 all practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, More conjectures on alternating sums of consecutive primes, a message to Number Theory List, March 2, 2013.

Cf. A008347, A005153, A208246, A213202.

f[n_]:=f[n]=FactorInteger[n]
Pow[n_,i_]:=Pow[n,i]=Part[Part[f[n],i],1]^(Part[Part[f[n],i],2])
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n],s+1],1]<=DivisorSigma[1,Product[Pow[n,i],{i,1,s}]]+1,0,1],{s,1,Length[f[n]]-1}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n,2]+Con[n]==0)
q[n_]:=q[n]=pr[n-4]==True&&pr[n]==True&&pr[n+4]==True
s[0_]:=0
s[n_]:=s[n]=Prime[n]-s[n-1]
a[n_]:=a[n]=Sum[If[n-s[m]>0&&q[n-s[m]],1,0],{m,1,n}]
Table[a[n],{n,1,100}]

A226743 Number of alternating sums of consecutive primes with result 2n-1.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 2, 3, 2, 3, 5, 4, 4, 4, 5, 4, 5, 5, 4, 6, 5, 4, 9, 5, 5, 7, 6, 6, 10, 7, 9, 5, 11, 6, 6, 9, 8, 8, 9, 9, 9, 12, 8, 8, 10, 7, 9, 9, 12, 11, 8, 11, 12, 6, 10, 6, 8, 14, 10, 12, 13, 10, 11, 5, 11, 9, 11, 16, 11, 11, 14, 10, 10, 13, 10, 17, 12, 11, 18, 13, 13, 11, 18, 11, 13, 12, 14, 16, 17, 14, 10, 15, 11, 12

Offset: 1

Ralf Stephan, Sep 01 2013

nonn

Since A008347 has no duplicate values, a(n) must be finite. This is not true for even results of the sum.

Sums of a single term are not included. - Robert Israel, Feb 06 2025

			n=5: 11-7+5=2*5-1, 13-11+7=2*5-1, 19-17+13-11+7-5+3=2*5-1, so a(5)=3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A084143.

N:= 100: M:= 2*N-1: # for a(1)..a(N)
p:= 1: b:= 0: B:= NULL:
for i from 1 do
  p:= nextprime(p);
  b:= b + (-1)^i*p;
  B:= B,b;
  if b > M then nB:= i; break fi;
od:
V:= Vector(M):
for j from 2 to nB by 2 while B[j] <= M do V[B[j]]:= 1 od:
for i from 1 to nB do
  for j from i+3 to nB by 2 do
    r:= abs(B[j]-B[i]);
    if r <= M then V[r]:= V[r]+1 else break fi;
od od:
seq(V[i],i=1..M,2); # Robert Israel, Feb 06 2025

vb=vecsmall(500);for(k=2,1000,forstep(l=k-1,1,-1,t=sum(i=l,k,prime(i)*(-1)^(k-i));if(t<500,vb[t]=vb[t]+1)))

A233809 a(n) = Sum_{k=1..n} prime(k) * s(k), where s(k) = (-1)^(floor(k/2)).

Original entry on oeis.org

2, -1, -6, 1, 12, -1, -18, 1, 24, -5, -36, 1, 42, -1, -48, 5, 64, 3, -64, 7, 80, 1, -82, 7, 104, 3, -100, 7, 116, 3, -124, 7, 144, 5, -144, 7, 164, 1, -166, 7, 186, 5, -186, 7, 204, 5, -206, 17, 244, 15, -218, 21, 262, 11, -246, 17, 286, 15, -262

Offset: 1

Jon Perry, Dec 16 2013

s(k) starts +1, -1, -1, +1, +1, -1, -1, ...

			a(6) = +2 - 3 - 5 + 7 + 11 - 13 = -1.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A008347, A233399.

Cf. A130642 (a(n) = -1), A130643 (a(n) = 1). - Michel Marcus, Aug 06 2017

[&+[NthPrime(k)*(-1)^(Floor(k/2)): k in [1..n]]: n in [1..60]]; // Vincenzo Librandi, Aug 07 2017

f[n_] := Sum[(-1)^Floor[k/2]*Prime[k], {k, n}]; Array[f, 60] (* Robert G. Wilson v, Aug 06 2017 *)

s(k) = (-1)^(floor(k/2));
a(n) = sum(k=1,n,s(k)*prime(k));
\\ Joerg Arndt, Aug 06 2017

Name corrected by Joerg Arndt, Aug 06 2017

A242188 a(n) = Sum_{i=1..n} (-1)^(i+1) prime(i)^3.

Original entry on oeis.org

0, 8, -19, 106, -237, 1094, -1103, 3810, -3049, 9118, -15271, 14520, -36133, 32788, -46719, 57104, -91773, 113606, -113375, 187388, -170523, 218494, -274545, 297242, -407727, 504946

Offset: 0

Timothy Varghese, May 22 2014

sign

For n even this is the negative of the sum of (3^3 - 2^3) + (7^3 - 5^3) + .. (prime(n)^3 - prime(n-1)^3). But this is half of the terms in the sum of (3^3 - 2^3) + (5^3 - 3^3) + (7^3 - 5^3) + ... + (prime(n)^3 - prime(n-1)^3) which has a sum that telescopes to prime(n)^3 - 8. Thus a good estimate of a(n) (half the terms) is prime(n)^3/2 (half the square of the n-th prime) which works well. For odd n, add prime(n)^2 to the estimate for even n.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A008347, A240860.

ListTools:-PartialSums([0,seq((-1)^(i+1)*ithprime(i)^3, i=1..40)]); # Robert Israel, Mar 09 2020

Table[Sum[(-1)^(i+1) Prime[i]^3,{i,n}],{n,0,30}] (* Harvey P. Dale, May 16 2021 *)

a(n) = sum(i=1, n, (-1)^(i+1)*prime(i)^3);

A339448 a(n) = (prime(n) - a(n-1)) mod 3; a(0)=0.

Original entry on oeis.org

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

Offset: 0

Simon Strandgaard, Dec 05 2020

nonn

			a(1) = ( 2 - 0) mod 3 = 2,
a(2) = ( 3 - 2) mod 3 = 1,
a(3) = ( 5 - 1) mod 3 = 1,
a(4) = ( 7 - 1) mod 3 = 0,
a(5) = (11 - 0) mod 3 = 2.

Cf. A008347.

a[0]=0; a[n_]:=Mod[Prime[n]-a[n-1],3]; Table[a[n],{n,0,85}] (* Stefano Spezia, Dec 05 2020 *)

require 'prime'
values = [0]
Prime.first(50).each do |prime|
    values << (prime-values[-1]) % 3
end
p values

A340867 a(n) = (prime(n) - a(n-1)) mod 4; a(0)=0.

Original entry on oeis.org

0, 2, 1, 0, 3, 0, 1, 0, 3, 0, 1, 2, 3, 2, 1, 2, 3, 0, 1, 2, 1, 0, 3, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 2, 1, 0, 3, 0, 1, 2, 3, 0, 1, 0, 3, 0, 3, 0, 1, 0, 3, 2, 1, 0, 3, 2, 1, 0, 1, 2, 3, 0, 3, 2, 3, 0, 1, 2, 3, 2, 1, 2, 3, 0, 3, 2, 3, 2, 3, 0, 1, 2, 3, 0

Offset: 0

Simon Strandgaard, Jan 24 2021

nonn

			a(1) = ( 2 - 0) mod 4 = 2,
a(2) = ( 3 - 2) mod 4 = 1,
a(3) = ( 5 - 1) mod 4 = 0,
a(4) = ( 7 - 0) mod 4 = 3,
a(5) = (11 - 3) mod 4 = 0.

Simon Strandgaard, Visualization

Cf. A008347, A339448.

a[0] = 0; a[n_] := a[n] = Mod[Prime[n] - a[n - 1], 4]; Array[a, 100, 0] (* Amiram Eldar, Jan 30 2021 *)

require 'prime'
values = [0]
Prime.first(50).each do |prime|
    values << (prime-values[-1]) % 4
end
p values

a(n) = A008347(n) mod 4.

A355726 a(n) = a(n-2) + prime(n-1) for a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 2, 3, 7, 10, 18, 23, 35, 42, 58, 71, 89, 108, 130, 151, 177, 204, 236, 265, 303, 336, 376, 415, 459, 504, 556, 605, 659, 712, 768, 825, 895, 956, 1032, 1095, 1181, 1246, 1338, 1409, 1505, 1582, 1684, 1763, 1875, 1956, 2072, 2155, 2283, 2378, 2510

Offset: 0

Paul Curtz, Jul 15 2022

nonn

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A077131 (even bisection), A077126 (odd bisection).

Cf. A008347 (first differences), (-1)^n*A330547 (second differences).

Join[{0},Accumulate[FoldList[#2-#1&,0,Prime[Range[100]]]]] (* Paolo Xausa, Dec 04 2023 *)

a(2*n) = A077131(n), for n>=1.

a(2*n+1) = A077126(n), for n>=1.

A376890 Alternating sum of twin primes (A001097).

Original entry on oeis.org

3, -2, 5, -6, 7, -10, 9, -20, 11, -30, 13, -46, 15, -56, 17, -84, 19, -88, 21, -116, 23, -126, 25, -154, 27, -164, 29, -168, 31, -196, 33, -206, 35, -234, 37, -244, 39, -272, 41, -306, 43, -376, 45, -386, 47, -414, 49, -472, 51, -518, 53, -546, 55, -562, 57, -584, 59, -600, 61, -748

Offset: 1

Ilya Gutkovskiy, Oct 08 2024

sign

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes.

Cf. A001097, A008347, A048598.

T1:= select(t -> isprime(t) and isprime(t+2), [seq(i,i=5..1000,6)]):
T:= map(t -> (-t, t+2), T1): T:= [3,op(T)]:
ListTools:-PartialSums(T); # Robert Israel, Nov 08 2024

a(n) = Sum_{k=1..n} (-1)^(k+1) * A001097(k).

a(2*n-1) = 2*n+1.

A370831 Alternating sum of composites.

Original entry on oeis.org

4, 2, 6, 3, 7, 5, 9, 6, 10, 8, 12, 9, 13, 11, 14, 12, 15, 13, 17, 15, 18, 16, 19, 17, 21, 18, 22, 20, 24, 21, 25, 23, 26, 24, 27, 25, 29, 26, 30, 27, 31, 29, 33, 30, 34, 31, 35, 33, 36, 34, 38, 36, 39, 37, 40, 38, 42, 39, 43, 41, 44, 42, 45, 43, 47, 44, 48, 45, 49, 46

Offset: 1

James C. McMahon, Mar 02 2024

nonn

Unlike equivalent sequence for primes, A008347, there are repeated terms.

			a(4) = 9 - 8 + 6 - 4 = 3.

Cf. A002808, A008347.

Join[{4},a[1]=4;a[n_]:=ResourceFunction["Composite"][n] - a[n-1];Table[a[n],{n,2,70}]] (* or with signs *) R=70;a[1]=4;a[n_]:=a[n-1]-ResourceFunction["Composite"][n] *(-1)^n;Table[a[n],{n,70}]

a(n) = A002808(n) - a(n-1), for n>1.

cp's OEIS Frontend

A175117 Numbers k such that Sum_(i=1..k) prime(i)*(-1)^(i+1) is a square.

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A213325 Number of ways to write n = q + sum_{k=1}^m(-1)^{m-k}p_k, where p_k is the k-th prime, and q is a practical number with q-4 and q+4 also practical.

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A226743 Number of alternating sums of consecutive primes with result 2n-1.

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A233809 a(n) = Sum_{k=1..n} prime(k) * s(k), where s(k) = (-1)^(floor(k/2)).

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A242188 a(n) = Sum_{i=1..n} (-1)^(i+1) prime(i)^3.

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A339448 a(n) = (prime(n) - a(n-1)) mod 3; a(0)=0.

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A340867 a(n) = (prime(n) - a(n-1)) mod 4; a(0)=0.

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A355726 a(n) = a(n-2) + prime(n-1) for a(0) = a(1) = 0.

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