A390950 - OEIS

%I #12 Dec 10 2025 06:51:02

%S 900,1764,1800,2700,3528,3600,4356,4500,4900,5292,5400,6084,7056,7200,

%T 8100,8712,9000,9800,10404,10584,10800,11025,12100,12168,12348,12996,

%U 13068,13500,14112,14400,15876,16200,16900,17424,18000,18252,19044,19600,20808,21168

%N Powerful numbers with more than 2 distinct prime factors.

%C A286708 is the disjoint union of this sequence and A355462.

%H Michael De Vlieger, <a href="/A390950/b390950.txt">Table of n, a(n) for n = 1..10000</a>

%F Intersection of A000977 and A001694.

%F Intersection of A286708 and A375055.

%F Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{p prime} 1/(p*(p-1)) - 1 - ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.012053688667932126846... . - _Amiram Eldar_, Dec 10 2025

%e Table of n, a(n) for select n:

%e  n     a(n)

%e -----------------------------------

%e  1     900 = 2^2 * 3^2 *  5^2

%e  2    1764 = 2^2 * 3^2 *  7^2

%e  3    1800 = 2^3 * 3^2 *  5^2

%e  4    2700 = 2^2 * 3^3 *  5^2

%e  5    3528 = 2^3 * 3^2 *  7^2

%e  6    3600 = 2^4 * 3^2 *  5^2

%e  7    4356 = 2^2 * 3^2 * 11^2

%e  8    4500 = 2^2 * 3^2 *  5^3

%e  9    4900 = 2^2 * 5^2 *  7^2

%e 10    5292 = 2^2 * 3^3 *  7^2

%e 22   11025 = 3^2 * 5^2 *  7^2

%e 79   44100 = 2^2 * 3^2 *  5^2 * 7^2

%t With[{nn = 25000}, Union@ Flatten@ Table[If[PrimeNu[#] > 2, #, Nothing] &[a^2*b^3], {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ] ]

%o (PARI) is_a390950(n) = ispowerful(n) && omega(n)>2 \\ _Hugo Pfoertner_, Dec 07 2025

%Y Cf. A000977, A001694, A286708, A355462, A375055.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Dec 01 2025