

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.


21



3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97
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OFFSET

1,1


COMMENTS

The first eight entries in A071561 but not in this sequence are 75, 78, 102, 105, 114, 138, 174 & 175.
The first eight entries in A239929 but not in this sequence are 21, 27, 33, 39, 51, 55, 57 & 65.
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 zth region of the symmetric spectrum of n has area a_z = 1/2 * (2^(m+1)  1) *(d_z + d_(2x+1z)), 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


LINKS



MATHEMATICA

(* path[n] and a237270[n] are defined in A237270 *)
atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], 1]  path[n1], 1]]]
Select[Range[100], atmostOneDiagonalsQ[#] && EvenQ[Length[a237270[#]]]&] (* data *)


CROSSREFS

Cf. A000203, A071561, A071562, A174905, A236104, A237270, A237271, A237593, A238443, A241010, A246955.


KEYWORD

nonn,more


AUTHOR



STATUS

approved



