A269931 Integers n such that the sum of squares of the first n primes (A024450) is the sum of 4 but no fewer nonzero squares.

4, 12, 20, 28, 29, 36, 44, 49, 52, 57, 60, 68, 73, 76, 84, 92, 100, 105, 108, 116, 124, 132, 140, 148, 153, 156, 161, 164, 172, 180, 188, 189, 196, 201, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 281, 284, 289, 292, 300, 308, 316, 324, 329, 332, 340, 345, 348, 356, 364, 372

Offset: 1

Author

Altug Alkan, Mar 08 2016

Keywords

• nonn

Comments

Terms that are not divisible by 4 are 29, 49, 57, 73, 105, 153, 161, 189, 201, 281, 289, 329, 345, 373, 385, 409, 417, 449, 457, 529, 553, 617, 633, 641, 645, ...

Corresponding values of sum of squares of the first n primes are 87, 4727, 30007, 98055, 109936, 239087, 486655, 710844, 874695, 1203356, 1432487, 2210983, 2841372, 3270831, ...

Examples

			4 is a term because 2^2 + 3^2 + 5^2 + 7^2 = 87 and 87 = x^2 + y^2 + z^2 has no solution for integer x, y and z.
5 is not a term because 2^2 + 3^2 + 5^2 + 7^2 + 11^2 = 208 = 8^2 + 12^2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004215, A024450.

Programs

Mathematica

Select[Range@ 372, Nand[SquaresR[4, #] > 1, Or[SquaresR[3, #] > 1, SquaresR[2, #] > 1, IntegerQ@ Sqrt@ #]] &@ Total[Prime[Range@ #]^2] &] (* Michael De Vlieger, Mar 08 2016 *)

PARI

isA004215(n)= my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri-7 ; if( j % 8==0, return(1) ) ; ); fouri *= 4 ; ) ; return(0) ;
a024450(n) = sum(k=1, n, prime(k)^2);
for(n=1, 1e3, if(isA004215(a024450(n)), print1(n, ", ")));

PARI

list(lim)=my(v=List(),n,s); forprime(p=2,, s+=p^2; if(n++>lim, return(Vec(v))); if(s\4^valuation(s, 4)%8==7, listput(v,n))) \\ Charles R Greathouse IV, Mar 08 2016