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A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

%I A027833 #38 Jan 20 2025 05:54:25
%S A027833 1,2,2,3,3,4,3,6,2,5,2,6,2,2,4,3,5,3,4,5,12,2,6,9,6,5,4,3,4,20,2,2,4,
%T A027833 4,19,2,3,2,4,8,11,5,3,3,3,10,5,4,2,17,3,6,3,3,9,9,2,6,2,6,5,6,2,3,2,
%U A027833 3,9,4,7,3,7,20,4,7,6,5,3,7,3,20,2,14,4,10,2,3,6,4,2,2,7,2,6,3
%N A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.
%C A027833 a(n) = number of primes p such that A014574(n) < p < A014574(n+1). - _Thomas Ordowski_, Jul 20 2012
%C A027833 Conjecture: a(n) < log(A014574(n))^2. - _Thomas Ordowski_, Jul 21 2012
%C A027833 Conjecture: All positive integers are represented in this sequence. This is verified up to 184, by searching up to prime indexes of ~128000000. The rate of filling-in the smallest remaining gap among the integers, and the growth in the maximum value found, both slow down considerably relative to a fixed quantity of twin prime incidences examined in each pass. The maximum value found was 237. - _Richard R. Forberg_, Jul 28 2016
%C A027833 All positive integers below 312 are in this sequence. - _Charles R Greathouse IV_, Aug 01 2016
%C A027833 From _Gus Wiseman_, Jun 11 2024: (Start)
%C A027833 Also the length of the n-th maximal antirun of prime numbers > 3, where an antirun is an interval of positions at which consecutive terms differ by more than 2. These begin:
%C A027833    5
%C A027833    7  11
%C A027833   13  17
%C A027833   19  23  29
%C A027833   31  37  41
%C A027833   43  47  53  59
%C A027833   61  67  71
%C A027833   73  79  83  89  97 101
%C A027833 (End)
%H A027833 Harvey P. Dale, <a href="/A027833/b027833.txt">Table of n, a(n) for n = 1..1000</a>
%H A027833 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%p A027833 A027833 := proc(n)
%p A027833     local plow,phigh ;
%p A027833     phigh := A001359(n+1) ;
%p A027833     plow := A001359(n) ;
%p A027833     numtheory[pi](phigh)-numtheory[pi](plow) ;
%p A027833 end proc:
%p A027833 seq(A027833(n),n=1..100) ; # _R. J. Mathar_, Jan 20 2025
%t A027833 Differences[Flatten[Position[Differences[Prime[Range[500]]],2]]] (* _Harvey P. Dale_, Nov 17 2018 *)
%t A027833 Length/@Split[Select[Range[4,10000],PrimeQ[#]&],#1+2!=#2&]//Most (* _Gus Wiseman_, Jun 11 2024 *)
%o A027833 (Sage)
%o A027833 def A027833(n) :
%o A027833    a = [ ]
%o A027833    st = 2
%o A027833    for i in (3..n) :
%o A027833       if (nth_prime(i+1)-nth_prime(i) == 2) :
%o A027833          a.append(i-st)
%o A027833          st = i
%o A027833    return(a)
%o A027833 A027833(496) # _Jani Melik_, May 15 2014
%o A027833 (PARI) n=1; p=5; forprime(q=7,1e3, if(q-p==2, print1(n", "); n=1, n++); p=q) \\ _Charles R Greathouse IV_, Aug 01 2016
%Y A027833 First differences of A029707 and A155752 = A029707 - 1. _M. F. Hasler_, Jul 24 2012
%Y A027833 Positions of first appearances are A373401, sorted A373402.
%Y A027833 Functional neighbors: A001359, A006512, A251092 or A175632, A373127 (firsts A373128, sorted A373200), A373403, A373405, A373409.
%Y A027833 A000040 lists the primes, differences A001223.
%Y A027833 A002808 lists the composite numbers, differences A073783.
%Y A027833 A046933 counts composite numbers between primes.
%Y A027833 A065855 counts composite numbers up to n.
%Y A027833 Cf. A025157, A038664, A071148,  A176246, A350842, A373406.
%K A027833 nonn
%O A027833 1,2
%A A027833 Jean-Marc MALASOMA (Malasoma(AT)entpe.fr)