A027833 - OEIS

A027833

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

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, 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, 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

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OFFSET

1,2

COMMENTS

a(n) = number of primes p such that A014574(n) < p < A014574(n+1). - Thomas Ordowski, Jul 20 2012

Conjecture: a(n) < log(A014574(n))^2. - Thomas Ordowski, Jul 21 2012

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

All positive integers below 312 are in this sequence. - Charles R Greathouse IV, Aug 01 2016

From Gus Wiseman, Jun 11 2024: (Start)

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:

7 11

13 17

19 23 29

31 37 41

43 47 53 59

61 67 71

73 79 83 89 97 101

(End)

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

MATHEMATICA

Differences[Flatten[Position[Differences[Prime[Range[500]]], 2]]] (* Harvey P. Dale, Nov 17 2018 *)

Length/@Split[Select[Range[4, 10000], PrimeQ[#]&], #1+2!=#2&]//Most (* Gus Wiseman, Jun 11 2024 *)

PROG

(Sage)

def A027833(n) :

a = [ ]

st = 2

for i in (3..n) :

if (nth_prime(i+1)-nth_prime(i) == 2) :

a.append(i-st)

st = i

return(a)

A027833(496) # Jani Melik, May 15 2014

(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

CROSSREFS

First differences of A029707 and A155752 = A029707 - 1. M. F. Hasler, Jul 24 2012

Positions of first appearances are A373401, sorted A373402.

Functional neighbors: A001359, A006512, A251092 or A175632, A373127 (firsts A373128, sorted A373200), A373403, A373405, A373409.

A000040 lists the primes, differences A001223.

A002808 lists the composite numbers, differences A073783.

A046933 counts composite numbers between primes.

A065855 counts composite numbers up to n.

Cf. A025157, A038664, A071148, A176246, A350842, A373406.

Sequence in context: A182921 A291268 A242767 * A110676 A117171 A325356

Adjacent sequences: A027830 A027831 A027832 * A027834 A027835 A027836

KEYWORD

nonn

AUTHOR

Jean-Marc MALASOMA (Malasoma(AT)entpe.fr)

STATUS

approved