cp's OEIS Frontend

A341677 Number of strictly inferior prime-power divisors of n.

%I A341677 #12 Nov 01 2024 05:14:21
%S A341677 0,0,0,0,0,1,0,1,0,1,0,2,0,1,1,1,0,2,0,2,1,1,0,3,0,1,1,2,0,3,0,2,1,1,
%T A341677 1,3,0,1,1,3,0,2,0,2,2,1,0,3,0,2,1,2,0,2,1,3,1,1,0,4,0,1,2,2,1,2,0,2,
%U A341677 1,3,0,4,0,1,2,2,1,2,0,4,1,1,0,4,1,1,1
%N A341677 Number of strictly inferior prime-power divisors of n.
%C A341677 We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.
%H A341677 Amiram Eldar, <a href="/A341677/b341677.txt">Table of n, a(n) for n = 1..10000</a>
%e A341677 The strictly inferior prime-power divisors of n!:
%e A341677 n = 1  2  6  24  120  720  5040  40320
%e A341677     ----------------------------------
%e A341677     .  .  2   2    2    2     2      2
%e A341677               3    3    3     3      3
%e A341677               4    4    4     4      4
%e A341677                    5    5     5      5
%e A341677                    8    8     7      7
%e A341677                         9     8      8
%e A341677                        16     9      9
%e A341677                              16     16
%e A341677                                     32
%e A341677                                     64
%e A341677                                    128
%t A341677 Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#<n/#&]],{n,100}]
%o A341677 (PARI) a(n) = sumdiv(n, d, d^2 < n && isprimepower(d)); \\ _Amiram Eldar_, Nov 01 2024
%Y A341677 Positions of zeros are A166684.
%Y A341677 The weakly inferior version is A333750.
%Y A341677 The version for odd instead of prime-power divisors is A333805.
%Y A341677 The version for prime instead of prime-power divisors is A333806.
%Y A341677 The weakly superior version is A341593.
%Y A341677 The version for squarefree instead of prime-power divisors is A341596.
%Y A341677 The strictly superior version is A341644.
%Y A341677 A000961 lists prime powers.
%Y A341677 A001221 counts prime divisors, with sum A001414.
%Y A341677 A001222 counts prime-power divisors.
%Y A341677 A005117 lists squarefree numbers.
%Y A341677 A038548 counts superior (or inferior) divisors.
%Y A341677 A056924 counts strictly superior (or strictly inferior) divisors.
%Y A341677 A207375 lists central divisors.
%Y A341677 - Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
%Y A341677 - Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341592, A341676.
%Y A341677 - Strictly Inferior: A060775, A070039, A341674.
%Y A341677 - Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341645, A341646, A341673.
%Y A341677 Cf. A000005, A000203, A000430, A001248, A006530, A020639.
%K A341677 nonn
%O A341677 1,12
%A A341677 _Gus Wiseman_, Feb 23 2021