A084143 -id:A084143 - cp's OEIS frontend

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

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)))

A282906 Expansion of 1/(1 - Sum_{j>=1} x^(Sum_{i=1..j} prime(i))).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 6, 5, 10, 9, 15, 18, 21, 32, 33, 52, 58, 79, 102, 122, 172, 201, 277, 341, 438, 575, 707, 947, 1169, 1530, 1949, 2474, 3228, 4046, 5281, 6678, 8594, 11035, 14025, 18142, 23015, 29681, 37888, 48512, 62319, 79456, 102230, 130456, 167418, 214356, 274221, 351904, 449700, 577024, 738150

Offset: 0

Ilya Gutkovskiy, Feb 24 2017

nonn

Number of compositions (ordered partitions) of n into partial sums of primes (A007504).

Conjecture: every number > 3 is the sum of at most 5 partial sums of primes.

			a(11) = 4 because we have [5, 2, 2, 2], [2, 5, 2, 2], [2, 2, 5, 2] and [2, 2, 2, 5].

Eric Weisstein's World of Mathematics, Prime Sums
Index entries for sequences related to compositions

Cf. A007504, A023360, A084143, A281273, A281274.

nmax = 60; CoefficientList[Series[1/(1 - Sum[x^Sum[Prime[i], {i, 1, j}], {j, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{j>=1} x^(Sum_{i=1..j} prime(i))).

A086834 Smallest prime that is the smallest of at least two consecutive primes with sum=n; a(n)=0 if no consecutive primes with sum=n exist.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 3, 0, 2, 0, 5, 0, 0, 3, 0, 2, 7, 0, 0, 0, 0, 5, 11, 0, 3, 0, 2, 0, 13, 7, 0, 0, 0, 0, 5, 0, 0, 3, 0, 2, 19, 0, 0, 0, 0, 0, 7, 13, 0, 0, 23, 5, 0, 0, 3, 0, 2, 17, 11, 0, 0, 0, 0, 0, 0, 7, 31, 0, 0, 19, 5, 0, 0, 3, 0, 2, 37, 0, 0, 0, 0, 11, 41, 0, 0, 0, 17, 0, 7, 0, 0, 0, 0, 5, 0

Offset: 1

Reinhard Zumkeller, Aug 08 2003

nonn

a(n) = 0 iff A084143(n) = 0.

			n = 72 = 5+7+11+13+17+19 = 13+17+19+23, as 5<13: a(72) = 5.

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

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A282906 Expansion of 1/(1 - Sum_{j>=1} x^(Sum_{i=1..j} prime(i))).

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A086834 Smallest prime that is the smallest of at least two consecutive primes with sum=n; a(n)=0 if no consecutive primes with sum=n exist.

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