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%I A341592 #18 Nov 01 2024 05:15:27
%S A341592 1,1,1,1,1,2,1,0,1,2,1,1,1,2,2,0,1,1,1,2,2,2,1,1,1,2,0,2,1,4,1,0,2,2,
%T A341592 2,1,1,2,2,1,1,4,1,2,1,2,1,0,1,1,2,2,1,0,2,1,2,2,1,3,1,2,1,0,2,4,1,2,
%U A341592 2,4,1,0,1,2,1,2,2,4,1,1,0,2,1,3,2,2,2
%N A341592 Number of squarefree superior divisors of n.
%C A341592 We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.
%H A341592 Amiram Eldar, <a href="/A341592/b341592.txt">Table of n, a(n) for n = 1..10000</a>
%e A341592 The strictly superior squarefree divisors (columns) of selected n:
%e A341592   1  6  8  30  60  210  420  630  1050  2310  4620  6930
%e A341592   ------------------------------------------------------
%e A341592   1  3  .   6  10   15   21   30   35     55    70   105
%e A341592      6     10  15   21   30   35   42     66    77   110
%e A341592            15  30   30   35   42   70     70   105   154
%e A341592            30       35   42   70  105     77   110   165
%e A341592                     42   70  105  210    105   154   210
%e A341592                     70  105  210         110   165   231
%e A341592                    105  210              154   210   330
%e A341592                    210                   165   231   385
%e A341592                                          210   330   462
%e A341592                                          231   385   770
%e A341592                                          330   462  1155
%e A341592                                          385   770  2310
%e A341592                                          462  1155
%e A341592                                          770  2310
%e A341592                                         1155
%e A341592                                         2310
%p A341592 with(numtheory):
%p A341592 a := n -> nops(select(d -> d*d >= n and issqrfree(d), divisors(n))):
%p A341592 seq(a(n), n = 1..88); # _Peter Luschny_, Feb 20 2021
%t A341592 Table[Length[Select[Divisors[n],SquareFreeQ[#]&&#>=n/#&]],{n,100}]
%o A341592 (PARI) a(n) = sumdiv(n, d, d^2 >= n && issquarefree(d)); \\ _Amiram Eldar_, Nov 01 2024
%Y A341592 Positions of zeros are A059172.
%Y A341592 The inferior version is A333749.
%Y A341592 The version for prime instead of squarefree divisors is A341591.
%Y A341592 The version for prime powers instead of squarefree divisors is A341593.
%Y A341592 The strictly superior case is A341595.
%Y A341592 The version for odd instead of squarefree divisors is A341675.
%Y A341592 A001221 counts prime divisors, with sum A001414.
%Y A341592 A033677 selects the smallest superior divisor.
%Y A341592 A038548 counts superior (or inferior) divisors.
%Y A341592 A056924 counts strictly superior (or strictly inferior) divisors.
%Y A341592 A161908 lists superior divisors.
%Y A341592 A207375 lists central divisors.
%Y A341592 - Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333750.
%Y A341592 - Superior: A051283, A063538, A063539, A070038, A116882, A116883, A341676.
%Y A341592 - Strictly Inferior: A060775, A333805, A333806, A341596, A341674.
%Y A341592 - Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341642, A341643, A341644, A341645, A341646, A341673.
%Y A341592 Cf. A000005, A000203, A001222, A001248, A006530, A020639, A112798.
%K A341592 nonn
%O A341592 1,6
%A A341592 _Gus Wiseman_, Feb 19 2021

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