cp's OEIS Frontend

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

%I A270713 #31 Jun 12 2025 14:11:05
%S A270713 1,2,225,4050,66528,113400,120960,92802153185280,
%T A270713 726046074908612178739200000000000,
%U A270713 3524292573661555639437312000000000000,2308850758786565168980497090478080000000000,142039354014714204088514497565910023710398021722450165760000000000000000
%N A270713 Numbers that are equal to the product of the number of divisors of their first k powers, for some k.
%C A270713 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
%C A270713 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
%H A270713 Max Alekseyev, <a href="/A270713/b270713.txt">Table of n, a(n) for n = 1..14</a>
%e A270713 d(4050) * d(4050^2) = 30 * 135 = 4050;
%e A270713 d(66528) * d(66528^2) = 96 * 693 = 66528.
%p A270713 with(numtheory): P:=proc(q) local a,k,n;
%p A270713 for n from 1 to q do a:=tau(n); k:=1;
%p A270713 while a<n do k:=k+1; a:=a*tau(n^k); od;
%p A270713 if n=a then print(n); fi; od; end: P(10^6);
%t A270713 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 *)
%o A270713 (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
%Y A270713 Cf. A000005, A270389.
%K A270713 nonn
%O A270713 1,2
%A A270713 _Paolo P. Lava_, Mar 22 2016
%E A270713 a(8)-a(10) from _Hiroaki Yamanouchi_, Apr 07 2016
%E A270713 a(11)-a(14) from _Max Alekseyev_, Jun 10 2025