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A114381 Sums of p-th to the q-th prime where p and q are consecutive primes. 5
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 (list; graph; refs; listen; history; text; internal format)
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
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 m-th to the n-th 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
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Feb 10 2006
STATUS
approved

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Last modified February 21 15:59 EST 2024. Contains 370237 sequences. (Running on oeis4.)