A184247 -id:A184247 - cp's OEIS frontend

A272863 Numerator of the ratio of consecutive prime gaps.

2, 1, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 1, 3, 2, 1, 3, 2, 3, 4, 1, 1, 2, 1, 2, 7, 2, 3, 1, 5, 1, 3, 1, 2, 3, 1, 1, 5, 1, 2, 1, 6, 1, 1, 1, 2, 3, 1, 5, 3, 1, 1, 1, 3, 2, 1, 5, 7, 2, 1, 2, 7, 3, 5, 1, 2, 3, 4, 3, 1, 2, 3, 4, 1, 2, 5, 1, 5, 1, 3, 2, 3, 4, 1, 1, 2, 3, 2, 1, 2, 1, 3, 2

Offset: 1

Andres Cicuttin, Jun 21 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A001223, A274225 (denominators), A274263.

Cf. A184247 (primes of the form prime(k+1) when a(k)=1).

[Numerator((NthPrime(n+2)-NthPrime(n+1))/(NthPrime(n+1)-NthPrime(n))): n in [1..100]]; // Vincenzo Librandi, Apr 27 2017

Table[(Prime[j+2]-Prime[j+1])/(Prime[j+1]-Prime[j]),{j,1,120}]//Numerator
Numerator[#[[2]]/#[[1]]]&/@Partition[Differences[Prime[Range[100]]],2,1] (* Harvey P. Dale, Dec 07 2022 *)

a(n) = numerator((prime(n+2)-prime(n+1))/(prime(n+1)-prime(n))); \\ Michel Marcus, Apr 29 2017

a(n) = numerator((prime(n+2)-prime(n+1))/(prime(n+1)-prime(n))).
A001223(n) = Product_{k=1..n-1} a(k)/A274225(k). - Andres Cicuttin, Apr 26 2017
A000040(n) = 3 + Sum_{j=1..n-1} Product_{k=1..j} a(k)/A274225(k), for n>1. - Andres Cicuttin, Apr 26 2017

A184248 Primes, q, such that for three consecutive primes, p, q & r, with p

Original entry on oeis.org

3, 5, 7, 13, 19, 31, 43, 53, 61, 73, 103, 109, 139, 151, 157, 173, 181, 193, 199, 211, 229, 241, 257, 263, 271, 283, 313, 349, 373, 401, 421, 433, 463, 467, 491, 509, 523, 563, 571, 593, 601, 607, 619, 643, 653, 661, 733, 743, 761, 811, 823, 829, 859

Offset: 1

Robert G. Wilson v, Jan 10 2011

The distance between the cited prime above to its immediate successor is divisible by the distance from that prime to its immediate predecessor.

Intersection(A184247, A184248): 5, 53, 157, 173, 211, .., = A006562: Balanced primes (of order 1).

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

Cf. A184247.

fQ[n_] := Block[{p = NextPrime[n, -1], q = n, r = NextPrime[n]}, IntegerQ[(r - q)/(q - p)]]; Select[ Prime@ Range@ 150, fQ]
Transpose[Select[Partition[Prime[Range[200]],3,1],IntegerQ[(#[[3]]- #[[2]])/ (#[[2]]-#[[1]])]&]][[2]] (* Harvey P. Dale, Mar 30 2014 *)

A179780 Primes, q, such that for three consecutive primes, p, q & r, with p

Original entry on oeis.org

23, 37, 47, 67, 79, 83, 89, 113, 127, 131, 163, 167, 233, 251, 277, 293, 307, 317, 331, 337, 353, 359, 367, 379, 383, 389, 409, 439, 443, 449, 479, 503, 547, 557, 577, 587, 613, 631, 647, 677, 683, 691, 701, 709, 719, 727, 739, 751, 757, 773, 787, 797, 839, 853, 863

Offset: 1

Robert G. Wilson v, Jan 10 2011

The distance between the cited prime above to its immediate predecessor and the distance from that prime to its immediate successor is a ratio a/b with neither a nor b equal to 1.

Complement(A000040, A184247 & A184248)

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

Cf. A184247 & A184248.

fQ[n_] := Block[{p = NextPrime[n, -1], q = n, r = NextPrime[n]}, !IntegerQ[(q - p)/(r - q)] && !IntegerQ[(r - q)/(q - p)]]; Select[ Prime@ Range@ 150, fQ]
Select[Partition[Prime[Range[200]],3,1],NoneTrue[{(#[[2]]-#[[1]])/ (#[[3]]- #[[2]]),(#[[3]]-#[[2]])/(#[[2]]-#[[1]])},IntegerQ]&][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 20 2016 *)

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A272863 Numerator of the ratio of consecutive prime gaps.

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A184248 Primes, q, such that for three consecutive primes, p, q & r, with p

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A179780 Primes, q, such that for three consecutive primes, p, q & r, with p

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