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A378984 Squares in A378769.

%I A378984 #30 May 11 2025 23:48:10
%S A378984 32400,63504,90000,129600,156816,202500,219024,254016,291600,345744,
%T A378984 360000,374544,467856,490000,518400,571536,627264,685584,777924,
%U A378984 810000,876096,960400,1016064,1089936,1166400,1210000,1245456,1382976,1411344,1440000,1498176,1587600
%N A378984 Squares in A378769.
%C A378984 Let omega = A001221, bigomega = A001222, and rad = A007947.
%C A378984 Numbers k that have all types of divisor pairs (d, k/d), d | k, that are listed in both A378769 and A378900. These are listed below:
%C A378984 Type A*: (Nontrivial) unitary divisor pairs, i.e., d coprime to k/d. The rest of the types are in cototient.
%C A378984 Type B*: gcd(d, k/d) > 1, rad(d) !| k/d, rad(k/d) !| d. These exist for k in A375055.
%C A378984 Type C: d < k/d, d | k/d but rad(k/d) !| d. Implies rad(k/d) = rad(k) and omega(d) < omega(k/d). These exist for k in A126706.
%C A378984 Type D: Either rad(d) | k/d, rad(k/d) !| d or vice versa. These exist for k in A378767.
%C A378984 Type E*: d = k/d = sqrt(k).
%C A378984 Type F: rad(d) = rad(k/d) = rad(k), d < k/d, d | k/d. These exist for k in A320966.
%C A378984 Type G*: rad(d) = rad(k/d) = rad(k), neither d | k/d nor k/d | d. These exist for k in A376936.
%C A378984 Asterisks denote symmetric types.
%C A378984 Since numbers d and k/d are either coprime or not, and if not, the squarefree kernel of one either divides the other or not, and if so, d divides k/d or not, and if so, d = k/d or not, there are no other types.
%C A378984 Smallest odd term is a(45) = 2480625.
%C A378984 Square roots not A350372: sqrt(810000) = 900 is not in A350372.
%H A378984 Michael De Vlieger, <a href="/A378984/b378984.txt">Table of n, a(n) for n = 1..10000</a>
%H A378984 Michael De Vlieger, <a href="/A378984/a378984.png">Diagram listing all divisor pairs for a(n)</a>, n = 1..8, showing type A in white, type B in light gray, type C in green or red, type D in blue or gold, type E in dark gray, type F in orange or purple, and type G in black.
%H A378984 Michael De Vlieger, <a href="/A378984/a378984_1.png">Diagram listing divisor pairs (d, k/d) for k = a(n)</a>, n = 1..60, showing only those with the smallest d and using the same color scheme as above, for each type and its reversal if the type is nonsymmetric.
%F A378984 This sequence is { k = s^2 : rad(k)^2 | k,
%F A378984   bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
%F A378984 Intersection of A378769 and A378900.
%F A378984 Intersection of A000290, A375055, and A376936.
%F A378984 Sum_{n>=1} = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} (1/(p^4-1))) - ((Sum_{p prime} (1/(p^2*(p^2-1))))^2 - Sum_{p prime} (1/(p^4*(p^2-1)^2)))/2 = 0.00015490158528995570146... . - _Amiram Eldar_, Dec 21 2024
%e A378984 a(1) = 32400 = 2^4 * 3^4 * 5^2 has the following divisor pair types:
%e A378984   Type A: 16 * 2025, Type B: 48 * 675, Type C: 2 * 16200, Type D: 8 * 4050
%e A378984   Type E: 180 * 180, Type F: 30 * 1080, Type G: 120 * 270.
%e A378984 a(2) = 63504 = 2^4 * 3^4 * 7^2 has the following divisor pair types:
%e A378984   Type A: 16 * 3969, Type B: 48 * 1323, Type C: 2 * 31752, Type D: 8 * 7938
%e A378984   Type E: 252 * 252, Type F: 42 * 1512, Type G: 168 * 378.
%e A378984 a(3) = 90000 = 2^4 * 3^2 * 5^4 has the following divisor pair types:
%e A378984   Type A: 9 * 10000, Type B: 18 * 5000, Type C: 2 * 45000, Type D: 8 * 11250
%e A378984   Type E: 300 * 300, Type F: 30 * 3000, Type G: 120 * 750, etc.
%t A378984 s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^21],  IntegerQ@ Sqrt[#] &];
%t A378984 t = Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
%t A378984 Select[t, PrimeOmega[#] > PrimeNu[#] > 2 &]
%Y A378984 Cf. A000290, A001221, A001222, A007947, A126706, A320966, A350372, A375055, A376936, A378767, A378769, A378900.
%Y A378984 Cf. A013661, A082020.
%K A378984 nonn,easy
%O A378984 1,1
%A A378984 _Michael De Vlieger_, Dec 15 2024