A335437 -id:A335437 - cp's OEIS frontend

A335438 Number of partitions of k_n into two distinct parts (s,t) such that k_n | s*t, where k_n = A335437(n).

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 2, 2, 1, 4, 1, 1, 3, 1, 4, 2, 1, 1, 5, 2, 1, 3, 1, 5, 3, 2, 1, 1, 4, 6, 1, 2, 1, 2, 1, 3, 6, 1, 4, 1, 1, 2, 1, 7, 1, 1, 5, 4, 3, 2, 2, 7, 1, 1, 1, 2, 1, 5, 8, 3, 1, 4, 1, 1, 1, 3, 8, 2, 1, 1, 6, 1, 3, 2, 1, 1, 2, 9, 5, 1, 1, 2, 1, 3

Offset: 1

Wesley Ivan Hurt, Jun 10 2020

nonn

a(n) >= 1.

			a(2) = 1; A335437(2) = 16 has exactly one partition into two distinct parts (12,4), such that 16 | 12*4 = 48. Therefore, a(2) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A000188, A013929, A335234, A335437.

f:= proc(n) local F,beta,t;
     F:= ifactors(n)[2];
     beta:= mul(t[1]^floor(t[2]/2),t=F);
     if beta <= 2 then NULL else floor((beta-1)/2) fi
end proc:
map(f, [$1..500]); # Robert Israel, Dec 23 2024

Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}] > 0, Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[(n - 1)/2]}], {}], {n, 400}] // Flatten

a(n) = floor((A000188(A335437(n))-1)/2). - Robert Israel, Dec 23 2024

A258211 Nonsquarefree numbers of the form 4*k + 2.

Original entry on oeis.org

18, 50, 54, 90, 98, 126, 150, 162, 198, 234, 242, 250, 270, 294, 306, 338, 342, 350, 378, 414, 450, 486, 490, 522, 550, 558, 578, 594, 630, 650, 666, 686, 702, 722, 726, 738, 750, 774, 810, 846, 850, 882, 918, 950, 954, 990, 1014, 1026, 1050, 1058, 1062, 1078, 1098, 1134, 1150

Offset: 1

Juri-Stepan Gerasimov, May 23 2015

The asymptotic density of this sequence is 1/4 - 2/Pi^2 = 0.047357... (A190357) - Amiram Eldar, Feb 10 2021
From Peter Munn, Jan 20 2022: (Start)
Even numbers whose square part is odd (and nontrivial).
If m is in the sequence then every odd multiple of m is in the sequence.
Closed under the operation A059896(.,.).
(End)

			18 is in this sequence because 4 * 4 + 2 = 18 = 2 * 3^2.

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

Intersection of A016825 and either A013929 or A335437.

Cf. A039956, A053850, A059896, A190357.

[n*4+2: n in [1..300] | not IsSquarefree(4*n+2)];

remove(numtheory:-issqrfree, [4*i+2 $ i=0..1000]); # Robert Israel, May 27 2015

Select [Range[300], ! SquareFreeQ[(4 # - 2)] &] 4 - 2 (* Vincenzo Librandi, May 24 2015 *)

select(n->!issquarefree(n), vector(50,n,2*n+9))*2 \\ Charles R Greathouse IV, May 26 2015

a(n) = 2*A053850(n). - Charles R Greathouse IV, May 26 2015

cp's OEIS Frontend

A335438 Number of partitions of k_n into two distinct parts (s,t) such that k_n | s*t, where k_n = A335437(n).

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