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%I #46 Jan 02 2023 12:30:47
%S 1,2,3,4,7,17,18,77
%N Numbers N such that {A028334(2), ..., A028334(K)} = {1,...,N} for some K >= 2, where A028334(k) = (prime(k+1) - prime(k))/2.
%C No further terms found using primes up to 10^12. - Douglas McNeil, Jan 14 2011
%C According to Thomas R. Nicely (see Links) the next term, if it exists, must correspond to a gap occurring between primes greater than 4*10^18. - _Giovanni Resta_, Jan 06 2013
%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]
%H Veikko Pohjola et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2011-January/006850.html">Differences of consecutive primes</a>, seqfan list, Jan 2011.
%e For k >= 2, consider A028334(k) = (1/2) * (prime(k+1) - prime(k)), half the k-th gap between primes. (We ignore g(1), which would equal 1/2.)
%e Then, using k=2,...,24 (and up to k=29), all the values 1,2,3 and 4 occur. Therefore, a(4)=4 is in the sequence.
%e However, for k=30 a new gap of 14 = 2*7 occurs, thus creating the "holes" (missing values) g=5 and g=6. The list of gaps has holes until one reaches k=46: At that moment all values g=1,...,7, and no other values occur. (This remains true up to k=98.) Therefore, a(5)=7 is in the list.
%e For more examples, see link to posts by Veikko Pohjola.
%o (PARI) p=2; L=l=g=1; while(p=nextprime(1+o=p), bittest(g,(p-o)\2) & next; a=(p-o)\2; g+=1<<a; a>L & L=a; l==a | next; while(bittest(g,l++),); l>L & print1( L ", "))
%Y Cf. A000230, A187779.
%K nonn
%O 1,2
%A _M. F. Hasler_, Jan 16 2011
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