cp's OEIS Frontend

A362866 Numbers k with the property that the parts of the symmetric representation of sigma(k) are two octagons.

%I A362866 #79 Sep 03 2023 10:52:20
%S A362866 10,14,22,26,34,38,46,58,62,74,82,86,94,106,118,122,134,142,146,158,
%T A362866 166,178,194,202,206,214,218,226,254,262,274,278,298,302,314,326,334,
%U A362866 346,358,362,382,386,394,398,422,446,454,458,466,478,482,502,514,526
%N A362866 Numbers k with the property that the parts of the symmetric representation of sigma(k) are two octagons.
%C A362866 Note that odd primes (A065091) are also the numbers j with the property that the parts of the symmetric representation of sigma(j) are two rectangles or more generally two quadrilaterals.
%C A362866 Conjecture 1: The octagons are S-shaped and they have width 1.
%C A362866 Conjecture 2: This sequence is also the primes doubled (or even semiprimes) >= 10 (Cf. A100484). - _Omar E. Pol_, Aug 15 2023
%C A362866 For the symmetric representation of sigma(n) to consist of 2 octagons the first 3 entries in row n of the triangle of A249223 must be nonzero, hence must be 1's, indicating width 1, with the remaining entries zero. Therefore, row n of A237048 is 100100..., implying n = 2*p with p>3 prime. Both conjectures are true. - _Hartmut F. W. Hoft_, Aug 22 2023
%C A362866 From _Omar E. Pol_, Aug 23 2023: (Start)
%C A362866 Also the row numbers of the triangle A364639 where the rows are [1, 0, -1, 1] or where the rows start with [1, 0, -1, 1] and the remaining terms are zeros.
%C A362866 Each supersequence A063221 >= 10 and A091999 >= 10 gives the numbers k with the property that the first part of the symmetric representation of sigma(k) is an  octagon. In that case each supersequence gives the row numbers of the triangle A364639 where the rows start with [1, 0, -1]. (End)
%e A362866 The symmetric representation of sigma(14) in the first quadrant is as follows:
%e A362866 .   _ _ _ _ _ _ _ _
%e A362866    |_ _ _ _ _ _ _  |
%e A362866                  | |
%e A362866                  | |_
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%e A362866                      |_ _
%e A362866                        | |_ _ _
%e A362866                        |_ _ _  |
%e A362866                              | |
%e A362866                              | |
%e A362866                              | |
%e A362866                              | |
%e A362866                              | |
%e A362866                              | |
%e A362866                              |_|
%e A362866 .
%e A362866 The diagram has only two parts (or polygons) and both are octagons so 14 is in the sequence.
%Y A362866 Subsequence of A063221, A091999, A100484, A239929 and possibly more.
%Y A362866 Cf. A000203, A001747, A065091, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A241008, A245092, A246955, A249223, A249351, A262626, A364414, A364639, A365081.
%K A362866 nonn
%O A362866 1,1
%A A362866 _Omar E. Pol_, May 06 2023
%E A362866 More terms from _Omar E. Pol_, Aug 15 2023