A127596 Numbers k such that 1 + Sum_{i=1..k-1} A001223(i)*(-1)^i = 0.

2, 4, 14, 22, 28, 233, 249, 261, 488, 497, 511, 515, 519, 526, 531, 534, 548, 562, 620, 633, 635, 2985, 3119, 3123, 3128, 3157, 4350, 4358, 4392, 4438, 4474, 4484, 4606, 4610, 4759, 5191, 12493, 1761067, 2785124, 2785152, 2785718, 2785729, 2867471

Offset

1,1

Comments

Or, with prime(0) = 1, numbers n such that Sum{k=0..n-1} (prime(k+1)-prime(k))*(-1)^k = Sum{k=0..n-1} (A008578(k+1)-A008578(k))*(-1)^k = 0.

There are 313 terms < 10^7, 846 terms < 10^8, 1161 terms < 10^9.

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1161 (first 846 terms from Klaus Brockhaus)

Eric Weisstein's World of Mathematics, Andrica's Conjecture

Eric Weisstein's World of Mathematics, Prime Difference Function

Example

1 - A001223(1) = 1 - 1 = 0, hence 2 is a term.
1 - A001223(1) + A001223(2) - A001223(3) = 1 - 1 + 2 - 2 = 0, hence 4 is a term.

Mathematica

S=0; Do[j=Prime[n+1]; i=Prime[n]; d[n]=j-i; S=S+(d[n]*(-1)^n); If[S+1==0, Print[Table[j|PrimePi[j]|S+1]]], {n, 1, 10^7, 1}]

Prog

(PARI) {m=10^8; n=1; p=1; e=1; s=0; while(n<m, q=nextprime(p+1); s=s+(q-p)*e; if(s==0, print1(n, ", ")); p=q; e=-e; n++)} /* Klaus Brockhaus, Apr 29 2007 */

Crossrefs

Cf. A001223 (differences between consecutive primes), A008578 (prime numbers at the beginning of the 20th century), A000101 (increasing gaps between primes, upper end), A002386 (increasing gaps between primes, lower end).

Cf. also A282178 (primes(a(n)), A330545, A330547.

Sequence in context: A008519 A243934 A071865 * A111871 A291079 A090808

Adjacent sequences: A127593 A127594 A127595 * A127597 A127598 A127599

Keyword

nonn

Author

Manuel Valdivia, Apr 03 2007

Extensions

Edited by Klaus Brockhaus, Apr 29 2007

Status

approved