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 A339245 #29 Feb 16 2025 08:34:01
%S A339245 216,360,504,600,792,864,936,1000,1080,1176,1224,1368,1400,1440,1512,
%T A339245 1656,1944,1960,2016,2088,2200,2232,2376,2400,2520,2600,2664,2744,
%U A339245 2808,2904,2952,3000,3096,3168,3240,3384,3400,3456,3672,3744,3800,3816,3960,4000,4056,4104,4200
%N A339245 Partrich numbers: positive integers whose square part and squarefree part are divisible by 2 and an odd prime.
%C A339245 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 A339245 Numbers whose odd part and even part are nonsquare and nonsquarefree.
%C A339245 All terms are divisible by 8. If m is present, 2m is absent and 4m is present.
%C A339245 Closed under multiplication by any square and under application of A059896: for n, k >= 1, A059896(a(n), k) is in the sequence.
%C A339245 From _Peter Munn_, Apr 07 2021: (Start)
%C A339245 The first deficient partrich number is 39304 = 2^3 * 17^3. (ascertained by _Amiram Eldar_)
%C A339245 The first 7 terms generate Carmichael numbers using the method of Erdős described in A287840.
%C A339245 (End)
%H A339245 Amiram Eldar, <a href="/A339245/b339245.txt">Table of n, a(n) for n = 1..10000</a>
%H A339245 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EvenPart.html">Even Part</a>, <a href="https://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 A339245 A008586 INTERSECT A028983 INTERSECT A036554 INTERSECT A038838.
%F A339245 Asymptotic density is 1/12 - 2/(3 * Pi^2) = 0.01578587757... . (Formula due to _Amiram Eldar_.)
%e A339245 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 A339245 n a(n)
%e A339245 1 216 = 2 * 3 * 4 * 9,
%e A339245 2 360 = 2 * 5 * 4 * 9,
%e A339245 3 504 = 2 * 7 * 4 * 9,
%e A339245 4 600 = 2 * 3 * 4 * 25,
%e A339245 5 792 = 2 * 11 * 4 * 9,
%e A339245 6 864 = 2 * 3 * 16 * 9,
%e A339245 7 936 = 2 * 13 * 4 * 9,
%e A339245 8 1000 = 2 * 5 * 4 * 25,
%e A339245 9 1080 = 2 * 15 * 4 * 9,
%e A339245 10 1176 = 2 * 3 * 4 * 49,
%e A339245 ...
%t A339245 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 A339245 Subsequence of A008586, A028983, A036554, A036785, A038838, A190892.
%Y A339245 Subsequences: A017139, A017643.
%Y A339245 Cf. A056832, A059896, A234957, A287840.
%K A339245 nonn,easy
%O A339245 1,1
%A A339245 _Peter Munn_, Nov 28 2020