

A114381


Sums of pth to the qth prime where p and q are consecutive primes.


4



8, 23, 41, 119, 109, 243, 187, 373, 689, 349, 991, 839, 551, 991, 1603, 1829, 841, 2155, 1717, 1079, 2689, 2081, 3113, 4359, 2641, 1667, 2867, 1779, 3037, 9905, 3627, 5293, 2357, 9125, 2599, 6265, 6593, 4889, 7081, 7327, 3219, 12253, 3487, 5933, 3631
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OFFSET

1,1


COMMENTS

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.
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


LINKS

Klaus Brockhaus, Table of n, a(n) for n=1..3245.


FORMULA

a(n) = Sum_{k=prime(n)..prime(n+1)} prime(k).  Danny Rorabaugh, Apr 01 2015


EXAMPLE

7 and 11 are consecutive primes. prime(7)+prime(8)+prime(9)+prime(10)+prime(11)= 119, the 4th entry in the table.


PROG

(PARI) g2(n)=for(x=1, n, print1(sumprimes(prime(x), prime(x+1))", "))
sumprimes(m, n) = /* Return the sum of the mth to the nth prime*/{ local(x); return(sum(x=m, n, prime(x))) }


CROSSREFS

Cf. A000040 (primes).
Cf. A161926 (numbers occurring at least twice), A161927 (index of second occurrence).  Klaus Brockhaus, Jul 01 2009
Sequence in context: A047719 A164131 A212458 * A139433 A226600 A178072
Adjacent sequences: A114378 A114379 A114380 * A114382 A114383 A114384


KEYWORD

easy,nonn


AUTHOR

Cino Hilliard, Feb 10 2006


STATUS

approved



