A391755 - OEIS

%I #10 Dec 24 2025 04:10:11

%S 27000,54000,74088,81000,108000,135000,148176,162000,216000,222264,

%T 243000,270000,287496,296352,324000,343000,405000,432000,444528,

%U 474552,486000,518616,540000,574992,592704,648000,666792,675000,686000,729000,810000,862488,864000,889056

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

%C Intersection of A036966 (cubefull numbers) and A000977 (numbers with more than 2 distinct prime factors).

%C Proper subset of A390950 which is the intersection of A001694 (powerful numbers) and A000977.

%H Michael De Vlieger, <a href="/A391755/b391755.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F Sum_{n>=1} 1/a(n) = Product_{p prime}(1 + 1/(p^2*(p-1))) - Sum_{p prime} 1/(p^2*(p-1)) - 1 - ((Sum_{p prime} (1/(p^2*(p-1))))^2 - Sum_{p prime} (1/(p^4*(p-1)^2)))/2 = 0.00023303003383679282415... . - _Amiram Eldar_, Dec 23 2025

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

%e  n        a(n)

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

%e   1     27000 = 2^3 * 3^3 * 5^3

%e   2     54000 = 2^4 * 3^3 * 5^3

%e   3     74088 = 2^3 * 3^3 * 7^3

%e   4     81000 = 2^3 * 3^4 * 5^3

%e   5    108000 = 2^5 * 3^3 * 5^3

%e   6    135000 = 2^3 * 3^3 * 5^4

%e   7    148176 = 2^4 * 3^3 * 7^3

%e   8    162000 = 2^4 * 3^4 * 5^3

%e   9    216000 = 2^6 * 3^3 * 5^3

%e  10    222264 = 2^3 * 3^4 * 7^3

%e  41   1157625 = 3^3 * 5^3 * 7^3

%e 161   9261000 = 2^3 * 3^3 * 5^3 * 7^3

%t nn = 2^20; s = Union@ Flatten@ Table[a^5*b^4*c^3, {c, Surd[nn, 3]}, {b, Surd[nn/(c^3), 4]}, {a, Surd[nn/(b^4*c^3), 5]}]; Select[s, PrimeNu[#] > 2 &]

%Y Cf. A000977, A001694, A036966, A126706, A286708, A390950.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Dec 22 2025