A078137 - OEIS

A078137

Numbers which can be written as sum of squares>1.

4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

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OFFSET

1,1

COMMENTS

A078134(a(n))>0.

Numbers which can be written as a sum of squares of primes. - Hieronymus Fischer, Nov 11 2007

Equivalently, numbers which can be written as a sum of squares of 2 and 3. Proof for numbers m>=24: if m=4*(k+6), k>=0, then m=(k+6)*2^2; if m=4*(k+6)+1 than m=(k+4)*2^2+3^2; if m=4*(k+6)+2 then m=(k+2)*2^2+2*3^2; if m=4*(k+6)+3 then m=k*2^2+3*3^2. Clearly, the numbers a(n)<24 can also be written as sums of squares of 2 and 3. Explicit representation as a sum of squares of 2 and 3 for numbers m>23: m=c*2^2+d*3^2, where c:=((floor(m/4) - 2*(m mod 4))>=0 and d:=m mod 4. - Hieronymus Fischer, Nov 11 2007

LINKS

Table of n, a(n) for n=1..70.

Index entries for sequences related to sums of squares

Eric Weisstein's World of Mathematics, Square Number.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

a(n)=n + 12 for n >= 12. - Hieronymus Fischer, Nov 11 2007

MATHEMATICA

Join[{4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22}, Range[24, 82]] (* Jean-François Alcover, Aug 01 2018 *)

PROG

(PARI) a(n)=if(n>11, n+12, [4, 8, 9, 12, 13, 16, 17, 18, 20, 21, 22][n]) \\ Charles R Greathouse IV, Aug 21 2011

CROSSREFS