A204099 -id:A204099 - cp's OEIS frontend

A204100 Number of integers between successive twin primes, divided by 3.

0, 1, 1, 3, 3, 5, 3, 9, 1, 9, 3, 9, 3, 1, 9, 3, 9, 3, 9, 11, 23, 3, 9, 19, 15, 9, 5, 7, 5, 49, 3, 1, 9, 7, 45, 3, 5, 3, 9, 19, 25, 15, 3, 3, 5, 35, 7, 9, 1, 39, 3, 15, 9, 7, 21, 27, 1, 17, 5, 15, 9, 17, 1, 7, 5, 3, 31, 9, 13, 9, 13, 55, 13, 21, 9, 7, 5, 19

Offset: 1

Michel Lagneau, Jan 10 2012

nonn

			a(2) = 1 because there exists three numbers 8, 9 and 10 between (5,7) and (11,13) => a(2) = 3/3 = 1.

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

Cf. A028334, A063091, A046933, A204099.

T:=array(1..100,1..2):k:=0:for n from 1 to 1000 do:p1:=ithprime(n):p2:=ithprime(n+1):if p2-p1 = 2 then k:=k+1:T[k,1]:=p1:T[k,2]:=p2:else fi:od: for p from 2 to k do:x:= T[p+1,1]- T[p,2]: printf(`%d, `,(x-1)/3):od:

Join[{0},Rest[(#[[2]]-#[[1]]-1)/3&/@Partition[Rest[Flatten[Select[ Partition[ Prime[Range[500]],2,1],#[[2]]-#[[1]]==2&]]],2]]] (* Harvey P. Dale, Jan 10 2016 *)

a(n) = (A063091(n+1)- A063091(n)-3)/3 = A204099(n)/3

A204136 Number of composites between successive twin prime pairs.

Original entry on oeis.org

0, 3, 3, 8, 8, 13, 8, 23, 3, 24, 9, 23, 9, 3, 25, 8, 24, 8, 25, 30, 59, 9, 23, 50, 41, 24, 13, 20, 13, 129, 9, 3, 25, 19, 118, 9, 14, 9, 25, 51, 66, 42, 8, 8, 14, 97, 18, 25, 3, 102, 8, 41, 26, 20, 56, 74, 3, 47, 15, 41, 24, 47, 3, 20, 15, 8, 86, 25, 34, 26

Offset: 1

Michel Lagneau, Jan 11 2012

nonn

			a(4)= 8 because between the 4th and 5th pairs of twins (17,19) and (29,31), there are 8 composites: 20, 21, 22, 24, 25, 26, 27, 28.

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

Cf. A048614, A204099.

T:=array(1..200,1..2):k:=0:for n from 1 to 1000 do:p1:=ithprime(n):p2:=ithprime(n+1):if p2-p1 = 2 then k:=k+1:T[k,1]:=p1:T[k,2]:=p2:else fi:od: for p from 1 to k do:i:= T[p,2]+1: j:= T[p+1,1]-1 :ii:=0:for q from i to j do:if type(q,prime)=false then ii:=ii+1:else fi:od: printf(`%d, `,ii):od:

nc[{a_,b_}]:=Count[Range[a+3,b-1],?(!PrimeQ[#]&)]; With[{tp=Partition[ Transpose[ Select[Partition[Prime[Range[ 500]],2,1],Last[#]-First[#] == 2&]][[1]],2,1]},nc/@tp] (* _Harvey P. Dale, Jun 25 2013 *)

a(n) = A204099(n) - A048614(n).

A213997 Number of integers between successive prime triples (p, p+2, p+6).

Original entry on oeis.org

0, 0, 17, 53, 0, 77, 29, 77, 29, 107, 173, 173, 29, 17, 203, 179, 17, 119, 47, 0, 113, 257, 119, 77, 149, 23, 383, 23, 557, 203, 59, 137, 239, 77, 119, 383, 113, 143, 137, 29, 257, 239, 17, 143, 539, 623, 1043, 203, 137, 53, 239, 317, 563, 23, 863, 89, 23, 707

Offset: 1

Michel Lagneau, Jun 30 2012

nonn

Property of this sequence: either a(n) is odd == 2 (mod 3), or a(n) =0.

			a(3)= 17 because between the 3rd and 4th prime triples there are 17 integers: (17,19,23), 24, 25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, (41,43,47).

Cf. A204099, A167132, A048614, A022004, A204100.

A213997 := proc(n)
        max(0,A022004(n+1)-A022004(n)-7) ;
end proc: # R. J. Mathar, Jul 11 2012

Flatten[Differences/@Table[Take[Flatten[Select[Partition[Prime[ Range[ 2000]],3,1], Differences[#]=={2,4}&]],{3n,3n+1}],{n,60}]]-1/.{-1->0} (* Harvey P. Dale, Jun 10 2014 *)

A242757 Partial sums of the number of integers between successive twin prime pairs.

Original entry on oeis.org

0, 3, 6, 15, 24, 39, 48, 75, 78, 105, 114, 141, 150, 153, 180, 189, 216, 225, 252, 285, 354, 363, 390, 447, 492, 519, 534, 555, 570, 717, 726, 729, 756, 777, 912, 921, 936, 945, 972, 1029, 1104, 1149, 1158, 1167

Offset: 1

Sam Mathers, Aug 16 2014

nonn

a(n) is the partial sum of the number of integers separating each successive pair of twin prime numbers less than and up to the n-th pair of twin primes.

			For n=4, a(4)=15 because the number of integers separating the first 4 pairs of twin prime numbers are as follows, 0 between (3,5) and (5,7), 3 between (5,7) and (11,13), 3 between (11,13) and (17,19), and 9 between (17,19) and (29,31). 0+3+3+9=15 so a(4)=15.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512. Partial sums of A204099.

s=0; q=2; forprime(p=5, 10^4, if(isprime(p+2), s=s+p-q-3; print1(s", "); q=p)) \\ Jens Kruse Andersen, Aug 17 2014

a(n) = A001359(n+1) - 2 - 3*n. - Robert Israel, Aug 17 2014

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A204100 Number of integers between successive twin primes, divided by 3.

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A204136 Number of composites between successive twin prime pairs.

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A213997 Number of integers between successive prime triples (p, p+2, p+6).

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A242757 Partial sums of the number of integers between successive twin prime pairs.

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