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

1,1

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000005, A211224, A271384.

Sequence in context: A191995 A168207 A082609 * A158787 A174706 A307269

Adjacent sequences: A271379 A271380 A271381 * A271383 A271384 A271385

Keyword

nonn,easy

Author

Paolo P. Lava, Apr 06 2016

Status

approved