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, 2308850758786565168980497090478080000000000, 142039354014714204088514497565910023710398021722450165760000000000000000

Offset: 1

Author

Paolo P. Lava, Mar 22 2016

Keywords

• nonn

Comments

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

The corresponding k are: 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5. - Michel Marcus, Apr 08 2016; updated by Max Alekseyev, Jun 11 2025

Examples

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

Links

• Max Alekseyev, Table of n, a(n) for n = 1..14

Crossrefs

Cf. A000005, A270389.

Programs

Maple

with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=tau(n); k:=1;
while a
				

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 *)

PARI

isok(m) = my(k = 1, prd = 1); while (prd < m, prd *= numdiv(m^k); k++); prd == m; \\ Michel Marcus, Apr 08 2016, Jun 12 2025

Extensions

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

a(11)-a(14) from Max Alekseyev, Jun 10 2025