A339245 Partrich numbers: positive integers whose square part and squarefree part are divisible by 2 and an odd prime.

216, 360, 504, 600, 792, 864, 936, 1000, 1080, 1176, 1224, 1368, 1400, 1440, 1512, 1656, 1944, 1960, 2016, 2088, 2200, 2232, 2376, 2400, 2520, 2600, 2664, 2744, 2808, 2904, 2952, 3000, 3096, 3168, 3240, 3384, 3400, 3456, 3672, 3744, 3800, 3816, 3960, 4000, 4056, 4104, 4200

Offset: 1

Author

Peter Munn, Nov 28 2020

Keywords

• nonn
• easy

Comments

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.

Numbers whose odd part and even part are nonsquare and nonsquarefree.

All terms are divisible by 8. If m is present, 2m is absent and 4m is present.

Closed under multiplication by any square and under application of A059896: for n, k >= 1, A059896(a(n), k) is in the sequence.

From Peter Munn, Apr 07 2021: (Start)

The first deficient partrich number is 39304 = 2^3 * 17^3. (ascertained by Amiram Eldar)

The first 7 terms generate Carmichael numbers using the method of Erdős described in A287840.

(End)

Examples

			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.
   n  a(n)
   1   216 = 2 *  3 *  4 *  9,
   2   360 = 2 *  5 *  4 *  9,
   3   504 = 2 *  7 *  4 *  9,
   4   600 = 2 *  3 *  4 * 25,
   5   792 = 2 * 11 *  4 *  9,
   6   864 = 2 *  3 * 16 *  9,
   7   936 = 2 * 13 *  4 *  9,
   8  1000 = 2 *  5 *  4 * 25,
   9  1080 = 2 * 15 *  4 *  9,
  10  1176 = 2 *  3 *  4 * 49,
  ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Even Part, Odd Part, Square Part, Squarefree Part.

Crossrefs

Subsequence of A008586, A028983, A036554, A036785, A038838, A190892.

Subsequences: A017139, A017643.

Cf. A056832, A059896, A234957, A287840.

Programs

Mathematica

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 *)

Formula

A008586 INTERSECT A028983 INTERSECT A036554 INTERSECT A038838.

Asymptotic density is 1/12 - 2/(3 * Pi^2) = 0.01578587757... . (Formula due to Amiram Eldar.)