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

10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 1

Author

Omar E. Pol, May 06 2023

Keywords

• nonn

Comments

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.

Conjecture 1: The octagons are S-shaped and they have width 1.

Conjecture 2: This sequence is also the primes doubled (or even semiprimes) >= 10 (Cf. A100484). - Omar E. Pol, Aug 15 2023

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

From Omar E. Pol, Aug 23 2023: (Start)

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.

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)

Examples

			The symmetric representation of sigma(14) in the first quadrant is as follows:
.   _ _ _ _ _ _ _ _
   |_ _ _ _ _ _ _  |
                 | |
                 | |_
                 |_ _|
                     |_ _
                       | |_ _ _
                       |_ _ _  |
                             | |
                             | |
                             | |
                             | |
                             | |
                             | |
                             |_|
.
The diagram has only two parts (or polygons) and both are octagons so 14 is in the sequence.
		

Crossrefs

Subsequence of A063221, A091999, A100484, A239929 and possibly more.

Cf. A000203, A001747, A065091, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A241008, A245092, A246955, A249223, A249351, A262626, A364414, A364639, A365081.

Extensions

More terms from Omar E. Pol, Aug 15 2023