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

1,9

Links

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

Example

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.

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

Prog

(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

Crossrefs

Cf. A055932, A346970.

Sequence in context: A317675 A332902 A204423 * A112170 A366475 A259976

Adjacent sequences: A376923 A376924 A376925 * A376927 A376928 A376929

Keyword

nonn

Author

Zhicheng Wei, Oct 10 2024

Status

approved