|
| |
|
|
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.
|
|
0
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
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=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
|
|
|
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 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Definition corrected, Mathematica duplicate removed, missing values added by R. J. Mathar, Jul 02 2009
|
|
|
STATUS
|
approved
|
| |
|
|
