cp's OEIS Frontend

A241008 Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is even, and that all parts have width 1.

%I A241008 #68 Oct 16 2023 13:42:38
%S A241008 3,5,7,10,11,13,14,17,19,21,22,23,26,27,29,31,33,34,37,38,39,41,43,44,
%T A241008 46,47,51,52,53,55,57,58,59,61,62,65,67,68,69,71,73,74,76,79,82,83,85,
%U A241008 86,87,89,92,93,94,95,97
%N A241008 Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is even, and that all parts have width 1.
%C A241008 The first eight entries in A071561 but not in this sequence are 75, 78, 102, 105, 114, 138, 174 & 175.
%C A241008 The first eight entries in A239929 but not in this sequence are 21, 27, 33, 39, 51, 55, 57 & 65.
%C A241008 The union of this sequence and A241010 equals A174905 (see link in A174905 for a proof). Updated by _Hartmut F. W. Hoft_, Jul 02 2015
%C A241008 Let n = 2^m * Product_{i=1..k} p_i^e_i = 2^m * q with m >= 0, k >= 0, 2 < p_1 < ... < p_k primes and e_i >= 1, for all 1 <= i <= k. For each number n in this sequence k > 0, at least one e_i is odd, and for any two odd divisors f < g of n, 2^(m+1) * f < g. Let the odd divisors of n be 1 = d_1 < ... < d_2x = q where 2x = sigma_0(q). The z-th region of the symmetric spectrum of n has area a_z = 1/2 * (2^(m+1) - 1) *(d_z + d_(2x+1-z)), for 1 <= z <= 2x. Therefore, the sum of the area of the regions equals sigma(n). For a proof see Theorem 6 in the link of A071561. - _Hartmut F. W. Hoft_, Sep 09 2015, Sep 04 2018
%C A241008 First differs from A071561 at a(43). - _Omar E. Pol_, Oct 06 2018
%H A241008 Hartmut F. W. Hoft, <a href="/A246955/a246955.pdf">Image for sigma(n) values</a>
%t A241008 (* path[n] and a237270[n] are defined in A237270 *)
%t A241008 atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
%t A241008 Select[Range[100], atmostOneDiagonalsQ[#] && EvenQ[Length[a237270[#]]]&] (* data *)
%Y A241008 Cf. A000203, A071561, A071562, A174905, A236104, A237270, A237271, A237593, A238443, A241010, A246955.
%K A241008 nonn,more
%O A241008 1,1
%A A241008 _Hartmut F. W. Hoft_, Aug 07 2014