A353282 a(n) is the number of solutions (x,y) to the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(n) when x >= y > 1 and y | x.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 1, 3, 1, 1, 2, 1, 3, 1, 2, 3, 1

Offset: 1

Author

Bernard Schott, Apr 09 2022

Keywords

• nonn

Comments

This is the generalization of a problem proposed by Yakov Perelman for A013929(93) = 243 (references, links and example).

a(n) is the number of squares > 1 dividing A013929(n), so, there is no solution (x,y) for S(x,y) = m when m is a squarefree number (A005117).

Also, number of times where A013929(n) appears in table A351381.

The smallest nonsquare number m such that equation S(x,y) = m has exactly n solutions, for n >= 0, is A130279(n+1).

Integers k for which number of solutions to the equation S(x,y) = k sets a new record are in A046952.

Examples

			For S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(2) = 8, the unique solution is (2,1) because (2+1) + (2-1) + (2*1) + (2/1) = 8, hence a(2) = 1.
For S(x,y) = A013929(93) = 243, the two solutions are (24,8) and (54,2) because S(24,8) = S(54,2) = 243, hence a(93) = 2 (problem from Perelman's book).
		

References

• I. Perelman, L'Algèbre récréative, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
• Ya. I. Perelman, Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.

Links

• Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
• Wikipedia, Yakov Perelman.

Crossrefs

Cf. A046951, A005117, A013929, A046952, A130279, A351381.

Programs

Mathematica

f[p_, e_] := 1 + Floor[e/2]; s[1] = 0; s[n_] := Times @@ (f @@@ FactorInteger[n]) - 1; s /@ Select[Range[250], ! SquareFreeQ[#] &] (* Amiram Eldar, Apr 09 2022 *)

Formula

a(n) = A046951(A013929(n)) - 1.

Extensions

More terms from Amiram Eldar, Apr 09 2022