A167401 a(n) is the smallest number k such that n*k has twice as many divisors as k.

1, 1, 2, 1, 12, 1, 4, 3, 20, 1, 72, 1, 28, 45, 8, 1, 108, 1, 160, 63, 44, 1, 288, 5, 52, 9, 224, 1, 10800, 1, 16, 99, 68, 175, 864, 1, 76, 117, 800, 1, 21168, 1, 352, 675, 92, 1, 1152, 7, 400, 153, 416, 1, 648, 275, 1568, 171, 116, 1, 259200

Offset: 2

Author

J. Lowell, Nov 02 2009

Keywords

• nonn
• look

Comments

a(n) is 1 for all prime numbers n.

From Robert Israel, Feb 09 2017: (Start)

All prime factors of a(n) divide n.

If n=p^k is a prime power, a(n) = p^(k-1).

If n=p*q with pA006881, a(n) = p^2*q. (End)

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A139315.

Programs

Maple

A167401 := proc(n) if isprime(n) then 1; else for a from 2 do if numtheory[tau](n*a) = 2*numtheory[tau](a) then return a ; end if; end do ; fi; end: seq(A167401(n),n=2..60) ; # R. J. Mathar, Nov 04 2009

Mathematica

tmd[n_]:=Module[{a=1},While[DivisorSigma[0,a*n]!=2DivisorSigma[0,a],a++];a]; Array[tmd,60,2] (* Harvey P. Dale, Apr 20 2013 *)

PARI

a(n) = {my(k=1); while (numdiv(n*k) != 2*numdiv(k), k++); k;} \\ Michel Marcus, Feb 10 2017

Extensions

Extended by Ray Chandler, Nov 10 2009

Extended beyond a(10) by R. J. Mathar, Nov 04 2009