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a(n) is the least multiplier k such that n*k has twice as many divisors as n.
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%I #37 Apr 23 2023 12:18:55

%S 2,3,2,3,2,4,2,3,2,3,2,5,2,3,2,3,2,4,2,3,2,3,2,5,2,3,2,3,2,4,2,3,2,3,

%T 2,5,2,3,2,3,2,4,2,3,2,3,2,5,2,3,2,3,2,4,2,3,2,3,2,6,2,3,2,3,2,4,2,3,

%U 2,3,2,5,2,3,2,3,2,4,2,3,2,3,2,5,2,3,2,3,2,4

%N a(n) is the least multiplier k such that n*k has twice as many divisors as n.

%C The zeros in A139315 are the missing values in this sequence (see A337709).

%C There are no 1's in this sequence. a(n) = 2 for all odd n and a(n) >= 3 for all even n. - _J. Lowell_, Sep 15 2020

%C Empirical observation: A007978(n) - a(n) = 1 for n = 60*A206547(n), = 2 for n = 420*A007310(n), else = 0. - _Hugo Pfoertner_, Sep 30 2020

%H Michel Marcus, <a href="/A337686/b337686.txt">Table of n, a(n) for n = 1..10000</a>

%H Hugo Pfoertner, <a href="/plot2a?name1=A007978&amp;name2=A337686&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawlines=true">Illustration of ratio A007978(n) / a(n)</a>, using Plot 2.

%F a(n) = A129902(n)/n.

%e a(1) = 2 because 1 has 1 divisor, 1*2 has 2 divisors, so 2 is the least multiplier to apply to 1 to get twice as many divisors.

%t nn = 105; Do[d[i] = DivisorSigma[0, i], {i, 12 nn}]; Reap[Do[m = 2; While[d[m i] != 2 d[i], m++]; Sow[m ], {i, nn}]][[-1, -1]]] (* _Michael De Vlieger_, Jan 10 2022 *)

%o (PARI) a(n) = {my(k=1); while (numdiv(n*k) != 2*numdiv(n), k++); k;}

%Y Cf. A000005, A129902, A139315, A337709 (missing values).

%K nonn

%O 1,1

%A _Michel Marcus_, Sep 15 2020