A116852 Number of partitions of n-th semiprime into 2 squares.

1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1

Offset: 1

Author

Jonathan Vos Post, Mar 15 2006

Keywords

• easy
• nonn

Comments

See also A000161 Number of partitions of n into 2 squares (when order does not matter and zero is allowed).

From Robert Israel, Jun 10 2020: (Start)

a(1)=1 if A001358(n) = p^2 where p is not in A002144.

a(n)=1 if A001358(n) = 2*p where p is in A002144.

a(n)=2 if A001358(n) = p*q where p and q are in A002144 (not necessarily distinct).

a(n)=0 otherwise. (End)

Examples

			a(1) = 1 because semiprime(1) = 4 = 0^2 + 2^2, the unique sum of squares.
a(2) = 0 because semiprime(2) = 6 has no decomposition into sum of 2 squares because it has a prime factor p == 3 (mod 4) with an odd power.
a(3) = 1 because semiprime(3) = 9 = 0^2 + 3^2, the unique sum of squares.
a(4) = 1 because semiprime(4) = 10 = 2*5 = 1^2 + 3^2.
a(9) = 2 because semiprime(9) = 25 = 0^2 + 5^2 = 3^2 + 4^2, two distinct ways.
a(23) = 2 because semiprime(23) = 65 = 5*13 = 1^2 + 8^2 = 4^2 + 7^2.
a(28) = 2 because semiprime(28) = 85 = 5*17 = 2^2 + 9^2 = 6^2 + 7^2.
a(49) = 2 because semiprime(49) = 145 = 5*29 = 1^2 + 12^2 = 8^2 + 9^2.
a(56) = 2 because semiprime(56) = 169 = 0^2 + 13^2 = 5^2 + 12^2.
a(60) = 2 because semiprime(60) = 185 = 5*37 = 4^2 + 13^2 = 8^2 + 11^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000161, A001358.

Programs

Maple

R:= NULL: count:= 0:
for n from 4 while count < 100 do
  if numtheory:-bigomega(n) = 2 then
    count:= count+1;
    F:= ifactors(n)[2];
    if nops(F) = 1 then
      if F[1][1] mod 4 = 1 then v:= 2
      else v:= 1
      fi
    elif F[1][1]=2 and F[2][1] mod 4 = 1 then v:= 1
    elif F[1][1] mod 4 = 1 and F[2][1] mod 4 = 1 then v:= 2
    else v:= 0
    fi;
    R:= R, v;
  fi
od:
R; # Robert Israel, Jun 10 2020

Formula

a(n) = A000161(A001358(n)).

Extensions

More terms from Giovanni Resta, Jun 15 2016