A376926 a(n) is the number of ways n can be written as x + y with x >= y, x and y coprime, and so that the distinct prime factors of x*y*n are consecutive primes starting with 2.

0, 1, 1, 1, 1, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 0, 0, 0, 0, 3, 1, 0, 0, 4, 0, 2, 1, 0, 0, 0, 1, 1, 0, 4, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Author

Zhicheng Wei, Oct 10 2024

Keywords

• nonn

Examples

			The a(25) = 4 solutions are:
  24 + 1 = 25 and 24 * 1 * 25 = 2^3 * 3 * 5^2;
  21 + 4 = 25 and 21 * 4 * 25 = 2^2 * 3 * 5^2 * 7;
  18 + 7 = 25 and 18 * 7 * 25 = 2 * 3^2 * 5^2 * 7;
  16 + 9 = 25 and 16 * 9 * 25 = 2^4 * 3^2 * 5^2.
The a(27) = 2 solutions are:
  25 + 2 = 27 and 25 * 2 * 27 = 2 * 3^3 * 5^2;
  20 + 7 = 27 and 20 * 7 * 27 = 2^2 * 3^3 * 5 * 7.
		

Links

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

Crossrefs

Cf. A055932, A346970.

Programs

Maple

f:= proc(n) local t,x,y,Pn,Px,Py,L;
   t:= 0:
   Pn:= numtheory:-factorset(n);
   for y from 1 to n/2 do
     x:= n-y;
     if igcd(x,y) > 1 then next fi;
     L:= Pn union numtheory:-factorset(x) union numtheory:-factorset(y);
     if max(L) = ithprime(nops(L)) then t:= t+1 fi
  od;
  t
end proc:
map(f, [$0..100]); # Robert Israel, Nov 12 2024

PARI

a(n)={sum(k=1, n\2, if(gcd(k,n-k)==1, my(f=factor(k*(n-k)*n)[,1]~); f[#f]==prime(#f)))} \\ Andrew Howroyd, Oct 12 2024