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A335437 Numbers k with a partition into two distinct parts (s,t) such that k | s*t. 2
9, 16, 18, 25, 27, 32, 36, 45, 48, 49, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 121, 125, 126, 128, 135, 144, 147, 150, 153, 160, 162, 169, 171, 175, 176, 180, 189, 192, 196, 198, 200, 207, 208, 216, 224, 225, 234, 240, 242, 243, 245, 250, 252, 256 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

All values of this sequence are nonsquarefree (A013929).

From Peter Munn, Nov 23 2020: (Start)

Numbers whose square part is greater than 4. [Proof follows from s and t having to be multiples of A019554(k), the smallest number whose square is divisible by k.]

Compare with A116451, numbers whose odd part is greater than 3. The self-inverse function A225546(.) maps the members of either one of these sets 1:1 onto the other set.

Compare with A028983, numbers whose squarefree part is greater than 2.

(End)

The asymptotic density of this sequence is 1 - 15/(2*Pi^2). - Amiram Eldar, Mar 08 2021

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Square part

Index entries for sequences related to partitions

EXAMPLE

16 is in the sequence since it has a partition into two distinct parts (12,4), such that 16 | 12*4 = 48.

MATHEMATICA

Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}] > 0, n, {}], {n, 300}] // Flatten

f[p_, e_] := p^(2*Floor[e/2]); sqpart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[256], sqpart[#] > 4 &] (* Amiram Eldar, Mar 08 2021 *)

CROSSREFS

Complement of A133466 within A013929.

Cf. A019554, A028983, A335234, A335438.

A038838, A046101, A062312\{1}, A195085 are subsequences.

Related to A116451 via A225546.

Sequence in context: A269563 A217570 A274188 * A034040 A048279 A250656

Adjacent sequences:  A335434 A335435 A335436 * A335438 A335439 A335440

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jun 10 2020

STATUS

approved

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Last modified October 25 22:20 EDT 2021. Contains 348256 sequences. (Running on oeis4.)