This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378769 #11 Dec 21 2024 11:09:42
%S A378769 5400,9000,10584,10800,13500,16200,18000,21168,21600,24696,26136,
%T A378769 27000,31752,32400,36000,36504,37044,40500,42336,43200,45000,48600,
%U A378769 49000,49392,52272,54000,62424,63504,64800,67500,68600,72000,73008,74088,77976,78408,81000,84672
%N A378769 Intersection of A375055 and A376936.
%C A378769 Let omega = A001221, bigomega = A001222, rad = A007947.
%C A378769 Powerful numbers k with bigomega(k) > omega(k) > 2 that are divisible by two distinct prime cubes p^3 and q^3.
%C A378769 Numbers k such that there exists (d, k/d), d | k, such that d neither divides nor is coprime to k/d and vice versa in the following 3 ways:
%C A378769 Type A: rad(d) does not divide d/k and rad(d/k) does not divide d
%C A378769 Type B: rad(d) divides d/k but rad(d/k) does not divide d
%C A378769 Type C: rad(d) | d/k and rad(d/k) | d, hence rad(d) = rad(d/k) = rad(k), a kind of coreful divisor pair.
%C A378769 Since (d, d/k) are noncoprime and do not divide one another, both must be composite, thus k is also composite.
%C A378769 In addition the following kinds of divisor pairs are also seen:
%C A378769 Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
%C A378769 Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.
%C A378769 A378767 = { k : omega(k) > 1, p^3 | k for some prime p }, and
%C A378769 A376936 = { k : rad(k)^2 | k, p^3 | k and q^3 | k for distinct primes p, q }.
%C A378769 Therefore, we need only take intersection of A375055 and A376936.
%H A378769 Michael De Vlieger, <a href="/A378769/b378769.txt">Table of n, a(n) for n = 1..10000</a>
%H A378769 Michael De Vlieger, <a href="/A378769/a378769.png">Listing of select divisor pairs of a(n)</a>, n = 1..16, showing divisor pairs of type A in light gray, type B in blue and gold, and type C in black.
%F A378769 Intersection of A375055, A376936, and A378767.
%F A378769 This sequence is { k : rad(k)^2 | k, bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
%F A378769 Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) - ((Sum_{p prime} (1/(p^2*(p-1))))^2 - Sum_{p prime} (1/(p^4*(p-1)^2)))/2 = 0.0025524144364532126894... . - _Amiram Eldar_, Dec 21 2024
%e A378769 Table of the first 12 terms of this sequence, showing examples of types A, B, and C described in Comments.
%e A378769 n a(n) Factors of a(n) Type A Type B Type C
%e A378769 ----------------------------------------------------------------
%e A378769 1 5400 2^3 * 3^3 * 5^2 24 * 225 4 * 1350 60 * 90
%e A378769 2 9000 2^3 * 3^2 * 5^3 18 * 500 4 * 2250 60 * 150
%e A378769 3 10584 2^3 * 3^3 * 7^2 24 * 441 4 * 2646 84 * 126
%e A378769 4 10800 2^4 * 3^3 * 5^2 48 * 225 8 * 1350 90 * 120
%e A378769 5 13500 2^2 * 3^3 * 5^3 12 * 1125 9 * 1500 90 * 150
%e A378769 6 16200 2^3 * 3^4 * 5^2 24 * 675 4 * 4050 60 * 270
%e A378769 7 18000 2^4 * 3^2 * 5^3 18 * 1000 8 * 2250 120 * 150
%e A378769 8 21168 2^4 * 3^3 * 7^2 48 * 441 8 * 2646 126 * 168
%e A378769 9 21600 2^5 * 3^3 * 5^2 50 * 432 8 * 2700 90 * 240
%e A378769 10 24696 2^3 * 3^2 * 7^3 18 * 1372 4 * 6174 84 * 294
%e A378769 11 26136 2^3 * 3^3 * 11^2 24 * 1089 4 * 6534 132 * 198
%e A378769 12 27000 2^3 * 3^3 * 5^3 24 * 1125 4 * 6750 60 * 450
%t A378769 s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^16],
%t A378769 Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
%t A378769 Select[s, PrimeOmega[#] > PrimeNu[#] > 2 &]
%Y A378769 Cf. A001694, A024619, A126706, A375055, A376936, A378767, A378900.
%Y A378769 Cf. A082020, A082695.
%K A378769 nonn,easy
%O A378769 1,1
%A A378769 _Michael De Vlieger_, Dec 13 2024