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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.

%I A376926 #74 Nov 13 2024 16:32:28
%S A376926 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,
%T A376926 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,
%U A376926 0,0,0,0,0,0,0,0,5,0,0,1,3,0,0,0,0,0,0,1,0,1
%N 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.
%H A376926 Robert Israel, <a href="/A376926/b376926.txt">Table of n, a(n) for n = 1..10000</a>
%e A376926 The a(25) = 4 solutions are:
%e A376926   24 + 1 = 25 and 24 * 1 * 25 = 2^3 * 3 * 5^2;
%e A376926   21 + 4 = 25 and 21 * 4 * 25 = 2^2 * 3 * 5^2 * 7;
%e A376926   18 + 7 = 25 and 18 * 7 * 25 = 2 * 3^2 * 5^2 * 7;
%e A376926   16 + 9 = 25 and 16 * 9 * 25 = 2^4 * 3^2 * 5^2.
%e A376926 The a(27) = 2 solutions are:
%e A376926   25 + 2 = 27 and 25 * 2 * 27 = 2 * 3^3 * 5^2;
%e A376926   20 + 7 = 27 and 20 * 7 * 27 = 2^2 * 3^3 * 5 * 7.
%p A376926 f:= proc(n) local t,x,y,Pn,Px,Py,L;
%p A376926    t:= 0:
%p A376926    Pn:= numtheory:-factorset(n);
%p A376926    for y from 1 to n/2 do
%p A376926      x:= n-y;
%p A376926      if igcd(x,y) > 1 then next fi;
%p A376926      L:= Pn union numtheory:-factorset(x) union numtheory:-factorset(y);
%p A376926      if max(L) = ithprime(nops(L)) then t:= t+1 fi
%p A376926   od;
%p A376926   t
%p A376926 end proc:
%p A376926 map(f, [$0..100]); # _Robert Israel_, Nov 12 2024
%o A376926 (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
%Y A376926 Cf. A055932, A346970.
%K A376926 nonn
%O A376926 1,9
%A A376926 _Zhicheng Wei_, Oct 10 2024