A138198 First occurrence of prime gaps which are squares.

2, 7, 1831, 9551, 89689, 396733, 11981443, 70396393, 1872851947, 10958687879, 47203303159, 767644374817, 8817792098461, 78610833115261, 497687231721157, 2069461000669981, 22790428875364879, 78944802602538877

Offset

0,1

Comments

More precisely, consider the possible squares which can occur as prime gaps: g_0=1, g_1=2^2, g_2=4^2, g_3=6^2, g_4=8^2, ... Then a(n) = smallest prime p(i) such that p(i+1)-p(i) = g_n, or a(n) = -1 if the gap g_n never occurs. - N. J. A. Sloane, Oct 28 2016

Links

Table of n, a(n) for n=0..17.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]

Example

Notes by Thomas R. Nicely:
No gap exceeding 1442 has been definitively established as a first occurrence; larger gaps included in these lists are instead first occurrences presently known of prime gaps. The smallest gap whose first occurrence remains uncertain is the (nonsquare) gap of 1208.
prime,gap
2, 1=1^2
7, 4=2^2
1831, 16=4^2
9551, 36=6^2
89689, 64=8^2
396733, 100=10^2
11981443, 144=12^2
70396393, 196=14^2
1872851947, 256=16^2
10958687879, 324=18^2
47203303159, 400=20^2
767644374817, 484=22^2
8817792098461, 576=24^2
78610833115261, 676=26^2
497687231721157, 784=28^2
2069461000669981, 900=30^2
22790428875364879, 1024=32^2
78944802602538877, 1156=34^2
2980374211158121907, 1296=36^2
18479982848279580912452968237, 1444=38^2
10338270318362067887873513954823823, 1600=40^2
5462539353768233509094313080601639583, 1764=42^2
9634432076725832064810529394509018411, 1936=44^2
24103660699017475735076387748469761375352177, 2116=46^2
1171872038536282864481405693168029955108099, (*48^2*)
169512938487733553802932479078305855585466971701227, (*50^2*)
228422210024736896126707605155690522381875250546666532046327, (*52^2*)
7229972437439469171089374324333535009566526827968927563, (*54^2*)
1263895714932859021916447978075625934206362807439043695674222113, (*56^2*)
569493611436727594340298806603382857255173440636060754222617328828425379, (*58^2*)
281376087412013738611508677824321032930454474305215907812114263492815921, (*60^2*)
680561565394793619717614472954048053005171290126070180152868857556290989645629867 (*62^2*)

Mathematica

Function[w, Prime@ First@ # & /@ Map[w[[ Key@ # ]] &, Select[Keys@ w, IntegerQ@ Sqrt@ # &]]]@ PositionIndex@ Differences@ Prime@ Range[10^7] (* Michael De Vlieger, Oct 27 2016 *)

Prog

(PARI) a(n)=my(k=max(1, 4*(n-1)^2), p=2); forprime(q=3, , if(q-p==k, return(p)); p=q) \\ Charles R Greathouse IV, Jun 05 2013

Crossrefs

Sequence in context: A000653 A128847 A123180 * A358482 A320505 A144836

Adjacent sequences: A138195 A138196 A138197 * A138199 A138200 A138201

Keyword

nonn

Author

Zak Seidov, Mar 05 2008

Extensions

Edited by N. J. A. Sloane, Oct 28 2016

Misprints in EXAMPLE fixed by Zak Seidov, Oct 18 2018

Status

approved