cp's OEIS Frontend

A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

%I A342087 #10 Nov 02 2024 09:14:39
%S A342087 1,2,2,2,2,4,2,4,2,4,2,6,2,4,4,4,2,6,2,6,4,4,2,8,2,4,4,6,2,8,2,6,4,4,
%T A342087 4,8,2,4,4,8,2,10,2,6,6,4,2,12,2,6,4,6,2,10,4,8,4,4,2,14,2,4,6,6,4,10,
%U A342087 2,6,4,8,2,16,2,4,6,6,4,10,2,12,4,4,2,14
%N A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.
%C A342087 An alternative wording: Number of chains of divisors starting with n and having all adjacent parts x > y^2.
%H A342087 Amiram Eldar, <a href="/A342087/b342087.txt">Table of n, a(n) for n = 1..10000</a>
%F A342087 For n > 1, a(n) = 2*A342083(n).
%e A342087 The chains for n = 1, 2, 6, 12, 24, 42, 48:
%e A342087    1    2      6        12        24        42          48
%e A342087         2/1    6/1      12/1      24/1      42/1        48/1
%e A342087                6/2      12/2      24/2      42/2        48/2
%e A342087                6/2/1    12/3      24/3      42/3        48/3
%e A342087                         12/2/1    24/4      42/6        48/4
%e A342087                         12/3/1    24/2/1    42/2/1      48/6
%e A342087                                   24/3/1    42/3/1      48/2/1
%e A342087                                   24/4/1    42/6/1      48/3/1
%e A342087                                             42/6/2      48/4/1
%e A342087                                             42/6/2/1    48/6/1
%e A342087                                                         48/6/2
%e A342087                                                         48/6/2/1
%t A342087 cem[n_]:=Prepend[Prepend[#,n]&/@Join@@cem/@Most[Divisors[n]],{n}];
%t A342087 Table[Length[Select[cem[n],And@@Thread[Divide@@@Partition[#,2,1]>Rest[#]]&]],{n,30}]
%Y A342087 The restriction to powers of 2 is A018819.
%Y A342087 Not requiring strict inferiority gives A067824.
%Y A342087 The weakly inferior version is twice A337135.
%Y A342087 The case ending with 1 is counted by A342083.
%Y A342087 The strictly superior version is A342084.
%Y A342087 The weakly superior version is A342085.
%Y A342087 The additive version is A342098, or A000929 allowing equality.
%Y A342087 A000005 counts divisors, with sum A000203.
%Y A342087 A001055 counts factorizations.
%Y A342087 A003238 counts chains of divisors summing to n-1, with strict case A122651.
%Y A342087 A038548 counts inferior (or superior) divisors.
%Y A342087 A056924 counts strictly inferior (or strictly superior) divisors.
%Y A342087 A067824 counts strict chains of divisors starting with n.
%Y A342087 A074206 counts ordered factorizations.
%Y A342087 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342087 A253249 counts strict chains of divisors.
%Y A342087 A334997 counts chains of divisors of n by length.
%Y A342087 Cf. A002033, A006530, A018819, A020639, A337105, A341674, A342086, A342094, A342095, A342097.
%K A342087 nonn
%O A342087 1,2
%A A342087 _Gus Wiseman_, Mar 05 2021