cp's OEIS Frontend

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

%I A385967 #10 Jul 20 2025 17:42:30
%S A385967 0,2,18,16,27,52,54,102,96,103,152,142,218,216,225,288,282,366,352,
%T A385967 387,440,474,558,528,559,648,626,758,780,783,900,858,978,976,1047,
%U A385967 1112,1146,1290,1248,1285,1404,1394,1550,1584,1587,1764,1710,1866,1868,1959,2048
%N A385967 Smallest nonnegative integer whose square is the sum of the squares of A047432(n) distinct primes.
%C A385967 Terms are odd when n is a multiple of 5, and even otherwise.
%C A385967 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.
%C A385967 From _David A. Corneth_, Jul 14 2025: (Start)
%C A385967 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.
%C A385967 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)
%e A385967 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.
%o A385967 (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]))}
%Y A385967 Cf. A047432, A018935.
%K A385967 nonn
%O A385967 1,2
%A A385967 _Charles L. Hohn_, Jul 13 2025