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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 (list; graph; refs; listen; history; text; internal format)
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.
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
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 *)
CROSSREFS
Sequence in context: A168252 A071561 A241561 * A335126 A367225 A173708
KEYWORD
nonn,more
AUTHOR
Hartmut F. W. Hoft, Aug 07 2014
STATUS
approved

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Last modified March 2 18:33 EST 2024. Contains 370498 sequences. (Running on oeis4.)