A333805 - OEIS

%I #25 Nov 02 2022 16:51:18

%S 0,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,2,1,1,2,1,1,2,1,1,2,1,1,3,1,1,2,1,

%T 2,2,1,1,2,2,1,2,1,1,3,1,1,2,1,2,2,1,1,2,2,2,2,1,1,3,1,1,3,1,2,2,1,1,

%U 2,3,1,2,1,1,3,1,2,2,1,2,2,1,1,3,2,1,2,1,1,4,2,1,2,1,2,2,1,2,3,2,1,2,1,1,4

%N Number of odd divisors of n that are < sqrt(n).

%C If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence counts strictly inferior odd divisors. - _Gus Wiseman_, Feb 26 2021

%H Antti Karttunen, <a href="/A333805/b333805.txt">Table of n, a(n) for n = 1..65537</a>

%F G.f.: Sum_{k>=1} x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

%e The strictly inferior odd divisors of 945 are 1, 3, 5, 7, 9, 15, 21, 27, so a(945) = 8. - _Gus Wiseman_, Feb 27 2021

%t Table[DivisorSum[n, 1 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 90}]

%t nmax = 90; CoefficientList[Series[Sum[x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) A333805(n) = sumdiv(n,d,(d%2)&&((d*d)<n)); \\ _Antti Karttunen_, Nov 02 2022

%Y Dominated by A001227 (number of odd divisors).

%Y Strictly inferior divisors (not just odd) are counted by A056924.

%Y The non-strict version is A069288.

%Y These divisors add up to A070039.

%Y The case of prime divisors is A333806.

%Y The strictly superior version is A341594.

%Y The case of squarefree divisors is A341596.

%Y The superior version is A341675.

%Y The case of prime-power divisors is A341677.

%Y A006530 selects the greatest prime factor.

%Y A020639 selects the smallest prime factor.

%Y - Odd -

%Y A000009 counts partitions into odd parts, ranked by A066208.

%Y A026424 lists numbers with odd Omega.

%Y A027193 counts odd-length partitions.

%Y A067659 counts strict partitions of odd length, ranked by A030059.

%Y - Inferior divisors -

%Y A033676 selects the greatest inferior divisor.

%Y A033677 selects the smallest superior divisor.

%Y A038548 counts superior (or inferior) divisors.

%Y A060775 selects the greatest strictly inferior divisor.

%Y A341674 lists strictly inferior divisors.

%Y Cf. A001248, A051283, A063538/A063539, A063962, A116883/A116882, A300272, A333807, A333809, A340832.

%K nonn

%O 1,12

%A _Ilya Gutkovskiy_, Apr 05 2020

%E Data section extended up to a(105) by _Antti Karttunen_, Nov 02 2022