A134755 Minimal number such that all greater numbers can be written as sums of squares of primes in more than n ways.

23, 39, 55, 64, 68, 73, 80, 84, 91, 96, 100, 105, 109, 113, 114, 118, 122, 123, 127, 131, 132, 136, 140, 140, 144, 145, 145, 149, 149, 153, 154, 156, 158, 160, 163, 164, 167, 168, 168, 172, 172, 176, 176, 176, 180, 180, 181, 181, 185, 185, 185, 189, 189, 190

Offset: 0

Author

Hieronymus Fischer, Nov 11 2007

Keywords

• nonn

Comments

The sequence is well-defined, in that a(n) exists for all n>=0. Proof by induction: a(0) exists. We set b(j):=number of ways to write j as sum of squares of primes (=A090677). If a(n) exists, then b(j)>n for all j>a(n). Setting m:=a(n)+1, we find that there are n+1 sum of squares of primes B(0,i), 1<=i<=n+1, with m=B(0,i).

Further there are n+1 such sum expressions B(1,i), B(2,i) and B(3,i), 1<=i<=n+1, representing m+1, m+2 and m+3, respectively. For all j>a(n) we have j=m+4*floor((j-m)/4)+(j-m) mod 4. Thus j=m+r+s*2^2, where r=0,1,2 or 3. Hence n can be written B(r,i)+s*2^2 and there are n+1 such representations.

Let q be the maximal prime number (to be squared) occurring as a term within those sum expressions B(r,i), 0<=r<=3,1<=i<=n+1. We select a prime number p>q and we set c:=a(n)+p^2. For j>c, we have the n+1 representations B(r(j),i)+s(j)*2^2. Additionally, for j-p^2 (which is >a(n)) there are also n+1 representations B(r_p,i)+s_p*2^2, where r_p:=r(j-p^2), s_p:=s(j-p^2).

Thus j can be written B(r(j),i)+s(j)*2^2, 1<=i<=n+1 and B(r_p,i)+s_p*2^2+p^2, 1<=i<=n+1. By choice of p all these sum representations of j are different, which implies, that there are 2n+2 such representations. It follows b(j)>2n+2>n+1 for all j>c, which implies, that a(n+1) exists.

Examples

			a(0)=23, since numbers >23 can be written as sum of squares of primes.
a(1)=39, since there are at least two ways, to write a number >39 as a sum of squares of primes.
		

Crossrefs

Cf. A078134, A078135, A078136, A078139, A090677, A078137, A134754.

Formula

a(n)=min( m | A090677(j)>n for all j>m).