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, 1799235198379903447, 30789586795090405813

Offset: 0

Author

Zak Seidov, Mar 05 2008

Keywords

• nonn

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

Examples

			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*)
		

Links

• Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
• The Prime Gap List, First occurrence prime gaps

Programs

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 *)

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

Formula

a(n) = A000230(2*n^2). - R. J. Mathar, Feb 13 2025

Extensions

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

Misprints in EXAMPLE fixed by Zak Seidov, Oct 18 2018

a(18)-a(19) from Brian Kehrig, May 02 2025