cp's OEIS Frontend

A239657 Number of odd divisors m of n such that there is a divisor d of n with d < m < 2*d.

%I A239657 #65 Jun 27 2025 17:27:08
%S A239657 0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,2,0,1,0,0,0,1,0,0,0,1,0,3,0,0,0,0,
%T A239657 1,2,0,0,0,1,0,3,0,0,3,0,0,1,0,0,0,0,0,3,0,1,0,0,0,3,0,0,1,0,0,3,0,0,
%U A239657 0,1,0,2,0,0,2,0,1,2,0,1,0,0,0,3,0,0,0,1,0,5,1,0,0,0,0,1,0,0,1,2,0,2,0,1,4,0,0,3,0,1,0,1,0,2,0,0,1,0,0,3,0,0,0,0,0,5,0,0
%N A239657 Number of odd divisors m of n such that there is a divisor d of n with d < m < 2*d.
%C A239657 The original name was: Number of odd divisors of n minus the number of parts in the symmetric representation of sigma(n).
%C A239657 Observation: at least the indices of the first 42 positive elements coincide with A005279: 6, 12, 15, 18, 20, 24..., checked (by hand) up to n = 2^7.
%C A239657 The observation is true for the indices of all positive elements. Hence the indices of the zeros give A174905. - _Omar E. Pol_, Jan 06 2017
%C A239657 a(n) is the number of subparts minus the number of parts in the symmetric representation of sigma(n). For the definition of "subpart" see A279387. - _Omar E. Pol_, Sep 26 2018
%C A239657 a(n) is the number of subparts of the symmetric representation of sigma(n) that are not in the first layer. - _Omar E. Pol_, Jan 26 2025
%H A239657 Antti Karttunen, <a href="/A239657/b239657.txt">Table of n, a(n) for n = 1..5000</a> (computed from the b-file of A237271 provided by Michel Marcus)
%F A239657 a(n) = A001227(n) - A237271(n).
%e A239657 Illustration of the symmetric representation of sigma(15) = 24 in the third quadrant:
%e A239657 .      _
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%e A239657 .     |_|_ _ _
%e A239657 .    8      | |_ _
%e A239657 .           |_    |
%e A239657 .             |_  |_
%e A239657 .            8  |_ _|
%e A239657 .                   |
%e A239657 .                   |_ _ _ _ _ _ _ _
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%e A239657 .                 8
%e A239657 .
%e A239657 For n = 15 the divisors of 15 are 1, 3, 5, 15, so the number of odd divisors of 15 is equal to 4. On the other hand the parts of the symmetric representation of sigma(15) are [8, 8, 8], there are three parts, so a(15) = 4 - 3 = 1.
%e A239657 From _Omar E. Pol_, Sep 26 2018: (Start)
%e A239657 Also the number of odd divisors of 15 equals the number of partitions of 15 into consecutive parts and equals the number of subparts in the symmetric representation of sigma(15). Then we have that the number of subparts minus the number of parts is  4 - 3 = 1, so a(15) = 1.
%e A239657 .      _
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%e A239657 .     |_|_ _ _
%e A239657 .    8      | |_ _
%e A239657 .           |_ _  |
%e A239657 .          7  |_| |_
%e A239657 .            1  |_ _|
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%e A239657 .                   |_ _ _ _ _ _ _ _
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%e A239657 .                 8
%e A239657 .
%e A239657 The above diagram shows the symmetric representation of sigma(15) with its four subparts: [8, 7, 1, 8]. (End)
%e A239657 From _Omar E. Pol_, Mar 30 2025: (Start)
%e A239657 The above diagram also shows that in the first layer there are three parts (having sizes [8, 7, 8]). Also there is another part that is not in the first layer, so a(15) = 1.
%e A239657 On the other hand for n = 15 there is only one odd divisor m of 15 such that  d < m < 2*d and d divides 15. That odd divisor is 5 as shown below, so a(15) = 1.
%e A239657    d  <  m  <  2*d
%e A239657 --------------------
%e A239657    1            2
%e A239657    3     5      6
%e A239657    5           10
%e A239657   15           30
%e A239657 .
%e A239657 For n = 18 there are two odd divisors m of 18 such that  d < m < 2*d and d divides 18. Those odd divisors are 3 and 9 as shown below, so a(18) = 2.
%e A239657    d  <  m  <  2*d
%e A239657 --------------------
%e A239657    1            2
%e A239657    2     3      4
%e A239657    3            6
%e A239657    6     9     12
%e A239657    9           18
%e A239657   18           36
%e A239657 .
%e A239657 (End)
%t A239657 a[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, Count[d, _?(OddQ[#[[2]]] && #[[2]] < 2*#[[1]] &)]]; Array[a, 100] (* _Amiram Eldar_, Apr 01 2025 *)
%Y A239657 Cf. A000203, A001227, A005279, A174905, A196020, A236104, A235791, A237048, A237270, A237271, A237591, A237593, A245092, A262626, A279387, A279391, A262626, A286000, A286001, A296508, A347186, A383147.
%K A239657 nonn
%O A239657 1,18
%A A239657 _Omar E. Pol_, Mar 23 2014
%E A239657 New Name from _Omar E. Pol_, Jan 26 2025