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%I #29 Feb 13 2021 14:37:06
%S 1,1,1,1,1,1,1,1,2,1,1,2,1,1,2,1,1,2,1,1,2,1,1,2,2,1,2,1,1,3,1,1,2,1,
%T 2,2,1,1,2,2,1,2,1,1,3,1,1,2,2,2,2,1,1,2,2,2,2,1,1,3,1,1,3,1,2,2,1,1,
%U 2,3,1,2,1,1,3,1,2,2,1,2,3,1,1,3,2,1,2,1,1,4
%N Number of odd divisors of n <= sqrt(n).
%C a(n) = #{d : d = A182469(n,k), d <= A000196(n), k=1..A001227(n)}. - _Reinhard Zumkeller_, Apr 05 2015
%H Reinhard Zumkeller, <a href="/A069288/b069288.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f.: Sum_{n>=1} 1/(1-q^(2*n-1)) * q^((2*n-1)^2). [_Joerg Arndt_, Mar 04 2010]
%e From _Gus Wiseman_, Feb 11 2021: (Start)
%e The inferior odd divisors for selected n are the columns below:
%e n: 1 9 30 90 225 315 630 945 1575 2835 4410 3465 8190 6930
%e --------------------------------------------------------------------
%e 1 3 5 9 15 15 21 27 35 45 63 55 65 77
%e 1 3 5 9 9 15 21 25 35 49 45 63 63
%e 1 3 5 7 9 15 21 27 45 35 45 55
%e 1 3 5 7 9 15 21 35 33 39 45
%e 1 3 5 7 9 15 21 21 35 35
%e 1 3 5 7 9 15 15 21 33
%e 1 3 5 7 9 11 15 21
%e 1 3 5 7 9 13 15
%e 1 3 5 7 9 11
%e 1 3 5 7 9
%e 1 3 5 7
%e 1 3 5
%e 1 3
%e 1
%e (End)
%t odn[n_]:=Count[Divisors[n],_?(OddQ[#]&&#<=Sqrt[n ]&)]; Array[odn,100] (* _Harvey P. Dale_, Nov 04 2017 *)
%o (PARI) a(n) = my(ir = sqrtint(n)); sumdiv(n, d, (d % 2) * (d <= ir)); \\ _Michel Marcus_, Jan 14 2014
%o (Haskell)
%o a069288 n = length $ takeWhile (<= a000196 n) $ a182469_row n
%o -- _Reinhard Zumkeller_, Apr 05 2015
%Y Cf. A000005, A000196, A001227, A069289, A182469.
%Y Positions of first appearances are A334853.
%Y A055396 selects the least prime index.
%Y A061395 selects the greatest prime index.
%Y - Odd -
%Y A000009 counts partitions into odd parts (A066208).
%Y A026424 lists numbers with odd Omega.
%Y A027193 counts odd-length partitions.
%Y A067659 counts strict partitions of odd length (A030059).
%Y - Inferior divisors -
%Y A033676 selects the greatest inferior divisor.
%Y A033677 selects the least superior divisor.
%Y A038548 counts inferior divisors.
%Y A060775 selects the greatest strictly inferior divisor.
%Y A063538 lists numbers with a superior prime divisor.
%Y A063539 lists numbers without a superior prime divisor.
%Y A063962 counts inferior prime divisors.
%Y A064052 lists numbers with a properly superior prime divisor.
%Y A140271 selects the least properly superior divisor.
%Y A217581 selects the greatest inferior divisor.
%Y A333806 counts strictly inferior prime divisors.
%Y Cf. A001055, A244991, A300272, A340101, A340607, A340832, A340854/A340855.
%K nonn
%O 1,9
%A _Reinhard Zumkeller_, Mar 14 2002
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