A391921 Powerful numbers that are not cubefree and are divisible by exactly 2 distinct primes.

72, 108, 144, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 675, 784, 800, 864, 968, 972, 1000, 1125, 1152, 1296, 1323, 1352, 1372, 1568, 1600, 1728, 1936, 1944, 2000, 2025, 2304, 2312, 2500, 2592, 2704, 2744, 2888, 2916, 3087, 3136, 3200, 3267, 3375, 3456

Offset

1,1

Comments

Numbers in A372404 divisible by 2 distinct prime factors.

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

Intersection of A007774 and A378767.

Intersection of A046099 and A355462.

Intersection of A355462 and A372404.

Intersection of A007774, A046099, and A286708.

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Formula

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

Example

Table of n, a(n) for select n:
 n   a(n)
--------------------
 1    72 = 2^3 * 3^2
 2   108 = 2^2 * 3^3
 3   144 = 2^4 * 3^2
 4   200 = 2^3 * 5^2
 5   216 = 2^3 * 3^3
 6   288 = 2^5 * 3^2
 7   324 = 2^2 * 3^4
 8   392 = 2^3 * 7^2
 9   400 = 2^4 * 5^2
10   432 = 2^4 * 3^3
11   500 = 2^2 * 5^3
14   675 = 3^3 * 5^2

Mathematica

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]] &]

Prog

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

Crossrefs

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

Sequence in context: A216426 A308053 A372404 * A356871 A359280 A395067

Adjacent sequences: A391918 A391919 A391920 * A391922 A391923 A391924

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 23 2025

Status

approved