%I #28 Apr 10 2021 22:27:37
%S 216,360,504,600,792,864,936,1000,1080,1176,1224,1368,1400,1440,1512,
%T 1656,1944,1960,2016,2088,2200,2232,2376,2400,2520,2600,2664,2744,
%U 2808,2904,2952,3000,3096,3168,3240,3384,3400,3456,3672,3744,3800,3816,3960,4000,4056,4104,4200
%N Partrich numbers: positive integers whose square part and squarefree part are divisible by 2 and an odd prime.
%C Not named after anyone, partrich numbers have the square part of their odd part, the square part of their even part (A234957), the squarefree part of their odd part and the squarefree part of their even part (A056832) all greater than 1.
%C Numbers whose odd part and even part are nonsquare and nonsquarefree.
%C All terms are divisible by 8. If m is present, 2m is absent and 4m is present.
%C Closed under multiplication by any square and under application of A059896: for n, k >= 1, A059896(a(n), k) is in the sequence.
%C From _Peter Munn_, Apr 07 2021: (Start)
%C The first deficient partrich number is 39304 = 2^3 * 17^3. (ascertained by _Amiram Eldar_)
%C The first 7 terms generate Carmichael numbers using the method of Erdős described in A287840.
%C (End)
%H Amiram Eldar, <a href="/A339245/b339245.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EvenPart.html">Even Part</a>, <a href="http://mathworld.wolfram.com/OddPart.html">Odd Part</a>, <a href="https://mathworld.wolfram.com/SquarePart.html">Square Part</a>, <a href="https://mathworld.wolfram.com/SquarefreePart.html">Squarefree Part</a>.
%F A008586 INTERSECT A028983 INTERSECT A036554 INTERSECT A038838.
%F Asymptotic density is 1/12 - 2/(3 * Pi^2) = 0.01578587757... . (Formula due to _Amiram Eldar_.)
%e A positive integer is present if and only if it factorizes as 2 times an odd squarefree number > 1, an even square that is a power of 4 and an odd square > 1. This factorization of the initial terms is shown below.
%e n a(n)
%e 1 216 = 2 * 3 * 4 * 9,
%e 2 360 = 2 * 5 * 4 * 9,
%e 3 504 = 2 * 7 * 4 * 9,
%e 4 600 = 2 * 3 * 4 * 25,
%e 5 792 = 2 * 11 * 4 * 9,
%e 6 864 = 2 * 3 * 16 * 9,
%e 7 936 = 2 * 13 * 4 * 9,
%e 8 1000 = 2 * 5 * 4 * 25,
%e 9 1080 = 2 * 15 * 4 * 9,
%e 10 1176 = 2 * 3 * 4 * 49,
%e ...
%t q[n_] := Module[{ie = IntegerExponent[n, 2], odd}, ie > 2 && OddQ[ie] && !SquareFreeQ[(odd = n/2^ie)] && !IntegerQ @ Sqrt[odd]]; Select[Range[4200], q] (* _Amiram Eldar_, Dec 04 2020 *)
%Y Subsequence of A008586, A028983, A036554, A036785, A038838, A190892.
%Y Subsequences: A017139, A017643.
%Y Cf. A056832, A059896, A234957, A287840.
%K nonn,easy
%O 1,1
%A _Peter Munn_, Nov 28 2020
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