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

1,1

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

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Hugo Pfoertner, Illustration of ratio A007978(n) / a(n), using Plot 2.

Formula

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A259940 A228829 A341982 * A007978 A245575 A333600

Adjacent sequences: A337683 A337684 A337685 * A337687 A337688 A337689

Keyword

nonn

Author

Michel Marcus, Sep 15 2020

Status

approved