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.

2, 14, 10, 26, 44, 45, 126, 68, 99, 104, 162, 117, 98, 124, 232, 164, 148, 200, 260, 333, 231, 244, 248, 297, 273, 284, 315, 406, 332, 345, 385, 430, 344, 399, 388, 436, 429, 488, 465, 495, 472, 525, 561, 555, 621, 556, 632, 604, 652, 712, 536, 693, 735, 675

Offset: 1

Author

Paolo P. Lava, Apr 06 2016

Keywords

• nonn
• easy

Examples

			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;
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.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..1200

Crossrefs

Cf. A000005, A211224, A271384.

Programs

Maple

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
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;
if a=h then print(n); break; fi; od; od; end: P(10^6);

Mathematica

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 *)

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;}
a(n) = {k=2; while (!isok(k, n), k++); k;} \\ Michel Marcus, Apr 07 2016