A369374
Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.
Original entry on oeis.org
36, 72, 108, 144, 216, 288, 324, 432, 576, 648, 864, 900, 972, 1152, 1296, 1728, 1800, 1944, 2304, 2592, 2700, 2916, 3456, 3600, 3888, 4500, 4608, 5184, 5400, 5832, 6912, 7200, 7776, 8100, 8748, 9000, 9216, 10368, 10800, 11664, 13500, 13824, 14400, 15552, 16200
Offset: 1
This sequence is the union of the following infinite sets:
P(2)^2 * A003586 = {36, 72, 108, 144, 216, 288, 324, ...}
= { m*P(2)^2 : rad(m) | P(2) }.
P(3)^2 * A051037 = {900, 1800, 2700, 3600, 4500, 5400, ...}
= { m*P(3)^2 : rad(m) | P(3) }.
P(4)^2 * A002473 = {44100, 88200, 132300, 176400, ...}
= { m*P(4)^2 : rad(m) | P(4) }, etc.
Cf.
A001221,
A001222,
A001694,
A002110,
A007947,
A055932,
A126706,
A286708,
A364930,
A367268,
A369417.
-
With[{nn = 2^14},
Select[
Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
Not@*PrimePowerQ],
And[EvenQ[#],
Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]
A369636
Powerful numbers k that are neither prime powers nor products of primorials.
Original entry on oeis.org
100, 108, 196, 200, 225, 324, 392, 400, 441, 484, 500, 648, 675, 676, 784, 800, 968, 972, 1000, 1089, 1125, 1156, 1225, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1764, 1936, 1944, 2000, 2025, 2116, 2312, 2500, 2601, 2700, 2704, 2744, 2888, 2916, 3025, 3087, 3136
Offset: 1
Let P(n) = A002110(n).
36 = 6^2 = P(2)^2 is a product of primorials and not in the sequence.
72 = 2 * 6^2 = P(1) * P(2)^2 is a product of primorials and not in the sequence.
a(1) = 100 = 2^2 * 5^2 is not a product of primorials.
a(2) = 108 = 2^2 * 3^3 is not a product of primorials, etc.
-
With[{nn = 3200}, Select[
Select[
Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
Not@*PrimePowerQ],
Nand[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1},
AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@
{#, FactorInteger[#]} &] ]
A380456
Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) < A006530(k), where omega = A001221.
Original entry on oeis.org
100, 196, 225, 400, 441, 484, 676, 784, 1000, 1089, 1156, 1225, 1444, 1521, 1600, 1764, 1936, 2025, 2116, 2500, 2601, 2704, 2744, 3025, 3136, 3249, 3364, 3375, 3844, 3969, 4225, 4356, 4624, 4761, 4900, 5476, 5625, 5776, 5929, 6084, 6400, 6724, 7056, 7225, 7396
Offset: 1
Table of n, a(n) for n = 1..12:
n a(n)
-----------------------------
1 100 = 10^2 = 2^2 * 5^2
2 196 = 14^2 = 2^2 * 7^2
3 225 = 15^2 = 3^2 * 5^2
4 400 = 20^2 = 2^4 * 5^2
5 441 = 21^2 = 3^2 * 7^2
6 484 = 22^2 = 2^2 * 11^2
7 676 = 26^2 = 2^2 * 13^2
8 784 = 28^2 = 2^4 * 7^2
9 1000 = 10^3 = 2^3 * 5^3
10 1089 = 33^2 = 3^2 * 11^2
11 1156 = 34^2 = 2^2 * 17^2
12 1225 = 35^2 = 5^2 * 7^2
Cf.
A001597,
A001694,
A002110,
A006530,
A053669,
A055932,
A080259,
A126706,
A131605,
A286708,
A369417,
A380446.
-
a053669[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q] ], q]; nn = 2^13; Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[a053669[#1] < #2[[-1, 1]], GCD @@ #2[[;; , -1]] > 1] & @@ {#, FactorInteger[#]} &]
Showing 1-3 of 3 results.
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