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 A378900 #17 Dec 21 2024 11:09:16
%S A378900 1296,5184,10000,11664,20736,32400,38416,40000,46656,50625,63504,
%T A378900 82944,90000,104976,129600,153664,156816,160000,186624,194481,202500,
%U A378900 219024,234256,250000,254016,291600,331776,345744,360000,374544,419904,455625,456976,467856,490000
%N A378900 Squares of numbers divisible by the squares of two distinct primes.
%C A378900 Also, the squares in A376936.
%C A378900 Proper subset of A378767, in turn a proper subset of A286708, the intersection of A001694 and A024619.
%C A378900 Numbers that have 3 kinds of coreful divisor pairs (d, k/d), d | k, i.e., rad(d) = rad(k/d) = rad(k) where rad = A007947. These kinds are described as follows:
%C A378900 Type A: d = k/d, which pertain to square k (in A000290).
%C A378900 Type B: d | k/d, d < k/d, which pertain to k in A320966, powerful numbers divisible by a cube.
%C A378900 Type C: neither d | k/d nor k/d | d, which pertain to k in A376936.
%C A378900 Since divisors d, k/d may either divide or not divide the other, there are no other cases.
%C A378900 In addition the following kinds of divisor pairs are also seen:
%C A378900 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 A378900 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.
%H A378900 Michael De Vlieger, <a href="/A378900/b378900.txt">Table of n, a(n) for n = 1..10000</a>
%H A378900 Michael De Vlieger, <a href="/A378900/a378900.png">Listing of select divisor pairs of a(n)</a>, n = 1..12, showing divisor pairs of type A in light gray, type B in orange and purple, and type C in black.
%F A378900 a(n) = A036785(n)^2.
%F A378900 Sum_{n>=1} 1/a(n) = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} 1/(p^4-1)) = 0.0015294876575980711757... . - _Amiram Eldar_, Dec 21 2024
%e A378900 Let b = A036785.
%e A378900 Table of the first 12 terms of this sequence, showing examples of types A, B, and C of coreful pairs of divisors.
%e A378900 n a(n) Factors of a(n) b(n) Type B Type C
%e A378900 -------------------------------------------------------------
%e A378900 1 1296 2^4 * 3^4 36 6 * 216 24 * 54
%e A378900 2 5184 2^6 * 3^4 72 6 * 864 48 * 108
%e A378900 3 10000 2^4 * 5^4 100 10 * 1000 40 * 250
%e A378900 4 11664 2^4 * 3^6 108 6 * 1944 24 * 486
%e A378900 5 20736 2^8 * 3^4 144 6 * 3456 54 * 384
%e A378900 6 32400 2^4 * 3^4 * 5^2 180 30 * 1080 120 * 270
%e A378900 7 38416 2^4 * 7^4 196 14 * 2744 56 * 686
%e A378900 8 40000 2^6 * 5^4 200 10 * 4000 80 * 500
%e A378900 9 46656 2^6 * 3^6 216 6 * 7776 48 * 972
%e A378900 10 50625 3^4 * 5^4 225 15 * 3375 135 * 375
%e A378900 11 63504 2^4 * 3^4 * 7^2 252 42 * 1512 168 * 378
%e A378900 12 82944 2^10 * 3^4 288 6 * 13824 54 * 1536
%t A378900 s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}], IntegerQ@ Sqrt[#] &] &[500000];
%t A378900 Union@ Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]
%Y A378900 Cf. A000290, A001694, A007947, A024619, A036785, A126706, A286708, A320966, A376936, A378768.
%Y A378900 Cf. A013661, A082020.
%K A378900 nonn,easy
%O A378900 1,1
%A A378900 _Michael De Vlieger_, Dec 12 2024