A391921 - OEIS

%I #12 Jan 06 2026 11:11:10

%S 72,108,144,200,216,288,324,392,400,432,500,576,648,675,784,800,864,

%T 968,972,1000,1125,1152,1296,1323,1352,1372,1568,1600,1728,1936,1944,

%U 2000,2025,2304,2312,2500,2592,2704,2744,2888,2916,3087,3136,3200,3267,3375,3456

%N Powerful numbers that are not cubefree and are divisible by exactly 2 distinct primes.

%C Numbers in A372404 divisible by 2 distinct prime factors.

%C A372404(30) = 1800 = 2^3 * 3^2 * 5^2 is not in this sequence.

%C Intersection of A007774 and A378767.

%C Intersection of A046099 and A355462.

%C Intersection of A355462 and A372404.

%C Intersection of A007774, A046099, and A286708.

%C Proper superset of A391896; this sequence also intersects A390539, e.g., A390539(1) = 72, but A390539(17) = 1800 = 2^3 * 3^2 * 5^2 is not in this sequence.

%H Michael De Vlieger, <a href="/A391921/b391921.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) = ((Sum_{p prime} 1/(p*(p-1)))^2 - Sum_{p prime} 1/(p*(p-1))^2)/2 - ((Sum_{p prime} 1/p^2)^2 - Sum_{p prime} 1/p^4)/2 = 0.094618784518255407065... . - _Amiram Eldar_, Jan 06 2026

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

%e  n   a(n)

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

%e  1    72 = 2^3 * 3^2

%e  2   108 = 2^2 * 3^3

%e  3   144 = 2^4 * 3^2

%e  4   200 = 2^3 * 5^2

%e  5   216 = 2^3 * 3^3

%e  6   288 = 2^5 * 3^2

%e  7   324 = 2^2 * 3^4

%e  8   392 = 2^3 * 7^2

%e  9   400 = 2^4 * 5^2

%e 10   432 = 2^4 * 3^3

%e 11   500 = 2^2 * 5^3

%e 14   675 = 3^3 * 5^2

%t nn = 3600; s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}]; Select[s, And[Length[#] == 2, AnyTrue[#, # > 2 &]] &@ FactorInteger[#][[;; , -1]] &]

%o (PARI) isok(k) = {my(e = factor(k)[, 2]); #e == 2 && vecmin(e) > 1 && vecmax(e) > 2;} \\ _Amiram Eldar_, Jan 06 2026

%Y Cf. A001694, A007774, A024619, A046099, A126706, A286708, A355462, A378767, A372404, A391896.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Dec 23 2025