A181687 Numbers k such that the number of odd divisors of (2k)^2 is an odd divisor of (2k)^2, and the number of even divisors of (2k)^2 is an even divisor of (2k)^2.

1, 2, 3, 6, 8, 12, 15, 21, 24, 25, 30, 33, 39, 42, 45, 50, 51, 57, 66, 69, 75, 78, 81, 87, 90, 93, 96, 102, 111, 114, 120, 123, 128, 129, 138, 141, 150, 159, 162, 168, 174, 177, 180, 183, 186, 189, 200, 201, 213, 219, 222, 225, 237, 240, 246, 249, 258, 264, 267, 282, 291, 300

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

A016742(a(n)) = A181795(n).

Maple

for j from 1 to 300 do
   y:=(2*j)^2:evdiv:=0:oddiv:=0:
   for k in divisors(y) do
      if(k mod 2=0)then evdiv:=evdiv+1:else oddiv:=oddiv+1:fi:
   od:
   if(type(y/evdiv, integer) and type(y/oddiv, integer))then
      print(j);
   fi;
od:

Mathematica

q[k_] := Module[{kk = 4*k^2, v = IntegerExponent[k, 2], m, e, no, ne}, m = k/2^v; e = If[m == 1, 0, FactorInteger[m][[;; , 2]]]; no = Times @@ (2*e + 1); ne = (2*v + 2)*no; OddQ[no] && Divisible[kk , no] && EvenQ[ne] && Divisible[kk , ne]]; Select[Range[300], q] (* Amiram Eldar, Mar 06 2026 *)

Prog

(PARI) isok(k) = {my(kk = 4*k^2, v = valuation(k, 2), m = k >> v, e = factor(m)[, 2], no, ne); no = vecprod(apply(x -> 2*x+1, e)); ne = (2*v+2)*no; (no % 2) && !(kk % no) && !(ne % 2) && !(kk % ne); } \\ Amiram Eldar, Mar 06 2026

Crossrefs

Cf. A016742, A181795.

Sequence in context: A198442 A035106 A122378 * A194881 A361922 A111242

Adjacent sequences: A181684 A181685 A181686 * A181688 A181689 A181690

Keyword

easy,nonn

Author

Nathaniel Johnston, Nov 17 2010

Status

approved