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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

A subsequence of A162164.

LINKS

Table of n, a(n) for n=1..42.

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

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=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

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

Sequence in context: A142389 A063350 A094492 * A062651 A142611 A283921

Adjacent sequences:  A162160 A162161 A162162 * A162164 A162165 A162166

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified June 16 14:56 EDT 2019. Contains 324152 sequences. (Running on oeis4.)