A270713 Numbers that are equal to the product of the number of divisors of their first k powers, for some k.

1, 2, 225, 4050, 66528, 113400, 120960, 92802153185280, 726046074908612178739200000000000, 3524292573661555639437312000000000000

Offset

1,2

Comments

a(2) = 2 is the only prime term possible, since the product of tau(p)^k is always even, and 2 is the only even prime. - Michael De Vlieger, Mar 27 2016

a(11) > 10^40. - Hiroaki Yamanouchi, Apr 07 2016

The corresponding k are: 1, 2, 3, 3, 3, 3, 3, 4, 5, 5. - Michel Marcus, Apr 08 2016

Links

Table of n, a(n) for n=1..10.

Example

d(4050) * d(4050^2) = 30 * 135 = 4050;
d(66528) * d(66528^2) = 96 * 693 = 66528.

Maple

with(numtheory): P:=proc(q) local a, k, n;
for n from 1 to q do a:=tau(n); k:=1;
while a<n do k:=k+1; a:=a*tau(n^k); od;
if n=a then print(n); fi; od; end: P(10^6);

Mathematica

Select[Insert[Complement[Range@ #, Prime@ Range@ PrimePi@ #] &[2 10^5], 2, 2], Function[k, AnyTrue[Range@ 3, Product[DivisorSigma[0, k^i], {i, #}] == k &]]] (* Michael De Vlieger, Mar 25 2016 *)

Prog

(PARI) isok(n) = {k = 1; prd = 1; while (prd < n, prd *= numdiv(n^k); k++); prd == n; } \\ Michel Marcus, Apr 08 2016

Crossrefs

Cf. A000005, A270389.

Sequence in context: A132936 A110715 A242835 * A071225 A212082 A015968

Adjacent sequences: A270710 A270711 A270712 * A270714 A270715 A270716

Keyword

nonn,more

Author

Paolo P. Lava, Mar 22 2016

Extensions

a(8)-a(10) from Hiroaki Yamanouchi, Apr 07 2016

Status

approved