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A271382 Least k with precisely n partitions k = x + y satisfying d(k) = d(x) + d(y), where d(k) is the number of divisors of k.

%I A271382 #13 Apr 09 2016 16:26:43
%S A271382 2,14,10,26,44,45,126,68,99,104,162,117,98,124,232,164,148,200,260,
%T A271382 333,231,244,248,297,273,284,315,406,332,345,385,430,344,399,388,436,
%U A271382 429,488,465,495,472,525,561,555,621,556,632,604,652,712,536,693,735,675
%N A271382 Least k with precisely n partitions k = x + y satisfying d(k) = d(x) + d(y), where d(k) is the number of divisors of k.
%H A271382 Paolo P. Lava, <a href="/A271382/b271382.txt">Table of n, a(n) for n = 1..1200</a>
%e A271382 d(10) = d(1) + d(9) = d(3) + d(7) = d(5) + d(5) = 4 and 10 is the least number with 3 partitions of two numbers with this property: therefore a(3) = 10;
%e A271382 d(126) = d(21) + d(105) = d(22) + d(104) = d(28) + d(98) = d(38) + d(33) = d(40) + d(86) = d(50) + d(76) = d(63) + d(63) = 12 and 126 is the least number with 7 partitions of two numbers with this property: therefore a(7) = 126.
%p A271382 with(numtheory): P:=proc(q) local a,h,k,n; for h from 1 to q do for n from 2*h to q do
%p A271382 a:=0; for k from 1 to trunc(n/2) do if tau(n)=tau(k)+tau(n-k) then a:=a+1; fi; od;
%p A271382 if a=h then print(n); break; fi; od; od; end: P(10^6);
%t A271382 nn = 10^3; Table[SelectFirst[Range@ nn, Function[k, With[{e = DivisorSigma[0, k]}, Count[Transpose@ {Range[k - 1, Ceiling[k/2], -1], Range@ Floor[k/2]}, x_ /; Total@ DivisorSigma[0, x] == e] == n]]], {n, 54}] (* _Michael De Vlieger_, Apr 06 2016 *)
%o A271382 (PARI) isok(k, n) = {my(nb = 0, tau = numdiv(k)); for (j=1, k\2, if (numdiv(j)+numdiv(k-j) == tau, nb++); if (nb > n, return (0));); nb == n;}
%o A271382 a(n) = {k=2; while (!isok(k, n), k++); k;} \\ _Michel Marcus_, Apr 07 2016
%Y A271382 Cf. A000005, A211224, A271384.
%K A271382 nonn,easy
%O A271382 1,1
%A A271382 _Paolo P. Lava_, Apr 06 2016