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A127596
Numbers k such that 1 + Sum_{i=1..k-1} A001223(i)*(-1)^i = 0.
6
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
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. A282178 (prime(a(n))), A330545, A330547.
Sequence in context: A008519 A243934 A071865 * A111871 A291079 A090808
KEYWORD
nonn
AUTHOR
Manuel Valdivia, Apr 03 2007
EXTENSIONS
Edited by Klaus Brockhaus, Apr 29 2007
STATUS
approved