A337686 a(n) is the least multiplier k such that n*k has twice as many divisors as n.

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, 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, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4

Offset: 1

Author

Michel Marcus, Sep 15 2020

Keywords

• nonn

Comments

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

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

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

Examples

			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.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000
• Hugo Pfoertner, Illustration of ratio A007978(n) / a(n), using Plot 2.

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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