A385967 Smallest nonnegative integer whose square is the sum of the squares of A047432(n) distinct primes.

0, 2, 18, 16, 27, 52, 54, 102, 96, 103, 152, 142, 218, 216, 225, 288, 282, 366, 352, 387, 440, 474, 558, 528, 559, 648, 626, 758, 780, 783, 900, 858, 978, 976, 1047, 1112, 1146, 1290, 1248, 1285, 1404, 1394, 1550, 1584, 1587, 1764, 1710, 1866, 1868, 1959, 2048

Offset: 1

Author

Charles L. Hohn, Jul 13 2025

Keywords

• nonn

Comments

Terms are odd when n is a multiple of 5, and even otherwise.

a(10) = 103 is also the smallest number whose square is the sum of the squares of at least 2 distinct primes and which itself is also a prime.

From David A. Corneth, Jul 14 2025: (Start)

If A047432(n) is a multiple of 4 then 4 cannot be one of the squares of primes in the sum. Proof: If 4 is there the sum of squares mod 4 will be 3 (mod 4) in that case. No square is 3 (mod 4). A contradiction.

If A047432(n) is a multiple of 3 then 9 cannot be one of the squares of primes in the sum. Proof: If 9 is there then the sum of squares will be 2 (mod 3) in that case. No square is 2 (mod 3). A contradiction. (End)

Examples

			a(5) = 27 because prime count A047432(5) = 6 and the smallest sum of squares of 6 distinct primes that is a square is 19^2 + 13^2 + 11^2 + 7^2 + 5^2 + 2^2 = 27^2.
		

Crossrefs

Cf. A047432, A018935.

Programs

PARI

a(n, c1=0, c2=0, c3=0, ~r, ~pc)={if(c1==0, n--; my(n5=n%5); n=(n-n5)/5*8+n5+if(n5>=2, 2, 0); r=[oo]; pc=vector(max(n-1, 0)); for(i=1, #pc, pc[i]=if(i>1, pc[i-1], 0)+prime(i)^2)); if(c1==n, return(if(issquare(c3), c3, oo))); for(i=n-c1, if(c1, c2-1, oo), my(p2=prime(i)^2); if(c3+p2+if(n-c1-1>0, pc[n-c1-1], 0)>=r[1], break); r[1]=min(r[1], a(n, c1+1, i, c3+p2, ~r, ~pc))); if(c1, r[1], sqrtint(r[1]))}