|
%I #18 Apr 01 2015 16:27:36
%S 8,23,41,119,109,243,187,373,689,349,991,839,551,991,1603,1829,841,
%T 2155,1717,1079,2689,2081,3113,4359,2641,1667,2867,1779,3037,9905,
%U 3627,5293,2357,9125,2599,6265,6593,4889,7081,7327,3219,12253,3487,5933,3631
%N Sums of p-th to the q-th prime where p and q are consecutive primes.
%C The number of terms in this sequence is infinite since there is no largest prime number. Conjecture: There will always be an n and i such that a(n) >= a(n+i) or the sequence will alternate forever. Equality does take place in the small sample shown with the entry 991. Certainly the proof of an infinity many twin primes would be a strong probable proof of this assertion. My guess is the alternation would always occur when a twin prime is encountered and often for other consecutive primes such as those differing by 4.
%C Some numbers occur (at least) twice: 991 at positions 11 and 14, 104435 at positions 193 and 348, 712363 at positions 654 and 2364. - _Klaus Brockhaus_, Jul 01 2009
%H Klaus Brockhaus, <a href="/A114381/b114381.txt">Table of n, a(n) for n=1..3245.</a>
%F a(n) = Sum_{k=prime(n)..prime(n+1)} prime(k). - _Danny Rorabaugh_, Apr 01 2015
%e 7 and 11 are consecutive primes. prime(7)+prime(8)+prime(9)+prime(10)+prime(11)= 119, the 4th entry in the table.
%o (PARI) g2(n)=for(x=1,n,print1(sumprimes(prime(x),prime(x+1))","))
%o sumprimes(m,n) = /* Return the sum of the m-th to the n-th prime*/{ local(x); return(sum(x=m,n,prime(x))) }
%Y Cf. A000040 (primes).
%Y Cf. A161926 (numbers occurring at least twice), A161927 (index of second occurrence). - _Klaus Brockhaus_, Jul 01 2009
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, Feb 10 2006
|
