A162163 Primes p such that p-1 and p+1 can individually be written as a sum of 2 and also as a sum of 3 distinct nonzero squares.

179, 467, 739, 809, 1097, 1171, 1619, 1801, 1873, 1907, 2467, 3203, 3331, 3491, 3923, 4051, 4177, 4211, 4931, 5507, 5651, 6067, 6121, 6353, 6569, 6659, 7219, 8081, 8243, 8297, 8353, 8819, 9091, 9161, 9377, 10243, 10531, 10657, 10729, 10889, 11251, 11699

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jun 26 2009, Jun 27 2009

Keywords

• nonn

Comments

A subsequence of A162164.

Examples

			p=12113: p-1=12112 = 36^2+40^2+96^2 = 36^2+104^2; p+1=12114 = 33^2+63^2+84^2 = 33^2+105^2.
p=4177: p-1=4176 = 24^2+60^2 = 24^2+36^2+48^2; p+1=4178 = 37^2+53^2 = 37^2+28^2+45^2. - _Vladimir Joseph Stephan Orlovsky_, Jun 26 2009
p=179: p-1=178 = 3^2+13^2 = 3^2+5^2+12^2; p+1=180 = 6^2+12^2=4^2+8^2+10^2. - _R. J. Mathar_, Jul 02 2009
		

Programs

Maple

isA004431 := proc(n) local x,y ; for x from 1 do if x^2 > n then RETURN(false); fi; y := n-x^2 ; if y> 0 and issqr(y ) then y := sqrt(y) ; if y <> x then RETURN(true) ; fi; fi; od: end:
isA004432 := proc(n) local x,y,z ; for x from 1 do if x^2 > n then RETURN(false); fi; for y from x+ 1 do if x^2+y^2>n then break ; fi; z := n-x^2-y^2 ; if z> 0 and issqr(z ) then z := sqrt(z) ; if z > y and z > x then RETURN(true) ; fi; fi; od: od: end:
for n from 1 to 2000 do p := ithprime(n) ; if isA004432(p-1) and isA004432(p+1) and isA004431(p-1) and isA004431(p+1) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jul 02 2009

Mathematica

f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++; If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[p=Prime[n];x=p-1;y=p+1;If[f[x]> 0&&f[y]>0,a=x-(f[x])^2;b=y-(f[y])^2;If[f[a]>0&&f[b]>0,c=(x-(f[x])^2-(f[a])^2)^(1/ 2);d=(y-(f[y])^2-(f[b])^2)^(1/2);If[c!=f[x]&&c!=f[a]&&f[x]!=f[a], If[d!=f[y]&&d!=f[b]&&f[y]!=f[b],AppendTo[lst,p]]]]],{n,3,6*6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 26 2009 *)

Formula

{p=A000040(i): p-1 in A004431 and p-1 in A004432 and p+1 in A004431 and p+1 in A004432}. - R. J. Mathar, Jul 02 2009

Extensions

Definition corrected, Mathematica duplicate removed, missing values added by R. J. Mathar, Jul 02 2009