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.

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

Offset: 1

Author

Hartmut F. W. Hoft, Aug 07 2014

Keywords

• nonn
• more

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.

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

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

First differs from A071561 at a(43). - Omar E. Pol, Oct 06 2018

Links

• Hartmut F. W. Hoft, Image for sigma(n) values

Crossrefs

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

Programs

Mathematica

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