%I #27 Apr 13 2022 07:44:35
%S 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,
%T 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,
%U 1,2,3,1,1,1,3,3,1,3,1,1,2,1,3,1,2,3,1
%N 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.
%C This is the generalization of a problem proposed by Yakov Perelman for A013929(93) = 243 (references, links and example).
%C 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).
%C Also, number of times where A013929(n) appears in table A351381.
%C The smallest nonsquare number m such that equation S(x,y) = m has exactly n solutions, for n >= 0, is A130279(n+1).
%C Integers k for which number of solutions to the equation S(x,y) = k sets a new record are in A046952.
%D I. Perelman, L'Algèbre récréative, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
%D Ya. I. Perelman, Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
%H Ya. I. Perelman, <a href="https://mirtitles.org/2012/05/23/yakov-perelman-algebra-can-be-fun/">Algebra Can Be Fun</a>, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Yakov_Perelman">Yakov Perelman</a>.
%F a(n) = A046951(A013929(n)) - 1.
%e 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.
%e 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).
%t 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 *)
%Y Cf. A046951, A005117, A013929, A046952, A130279, A351381.
%K nonn
%O 1,5
%A _Bernard Schott_, Apr 09 2022
%E More terms from _Amiram Eldar_, Apr 09 2022