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%I A262259 #54 Mar 10 2023 09:06:02
%S A262259 3,10,78,136,666,820,1830,2628,4656,5886,6328,16290,18528,28920,32896,
%T A262259 39340,48828,56616,62128,78606,80200,83436,88410,93528,100576,104196,
%U A262259 135460,146070,166176,180300,187578,190036
%N A262259 Numbers k such that the symmetric representation of sigma(k) has only two parts and they meet at the center of the Dyck path.
%C A262259 For a proof of the formula see the link and also the links in A239929 and A071561. This formula allows for a fast computation of numbers in the sequence that does not require computations of Dyck paths.
%C A262259 Subsequence of A239929.
%C A262259 A191363 is a subsequence.
%C A262259 All terms are triangular numbers.
%C A262259 More precisely, all terms are second hexagonal numbers (A014105). There are no terms with middle divisors. - _Omar E. Pol_, Oct 31 2018
%C A262259 Numbers k such that the concatenation of the widths of the symmetric representation of sigma(k) is a cyclops numbers (A134808). - _Omar E. Pol_, Aug 29 2021
%H A262259 Hartmut F. W. Hoft, <a href="/A262259/a262259.pdf">proof of formula</a>
%F A262259 Terms are equal to q*(2*q + 1) where q is in A174973 and 2*q + 1 is prime.
%e A262259 q = 128 = 2^7 is the 15th term in A174973 for which 2*n+1 = 2^8 + 1 is prime so that a(15) = 2^7 * (2^8 + 1) = 32896. The two parts in the symmetric representation of sigma of a(15) have width 1 and sigma(a(15)) = 2 * a(15) - 2.
%e A262259 q = 308 is the 32nd term in A174973 for which 2*n+1 is prime so that a(32) = 308 * 617 = 190036. The maximum width of the two regions is 2 and sigma(a(32)) = 415296.
%e A262259 For n = 21, the symmetric representation of sigma(21) has two parts that meet at the center of the Dyck path, but 21 is not in the sequence because the symmetric representation of sigma(21) has more than two parts. - _Omar E. Pol_, Sep 18 2015
%e A262259 From _Omar E. Pol_, Oct 05 2015: (Start)
%e A262259 Illustration of initial terms (n = 1, 2):
%e A262259 . y
%e A262259 .  |
%e A262259 .  |_ _ _ _ _ _
%e A262259 .  |_ _ _ _ _  |
%e A262259 .  |         | |_
%e A262259 .  |         |_ _|_
%e A262259 .  |             | |_ _
%e A262259 .  |             |_ _  |
%e A262259 .  |                 | |
%e A262259 .  |_ _              | |
%e A262259 .  |_ _|_            | |
%e A262259 .  |   | |           | |
%e A262259 .  |_ _|_|_ _ _ _ _ _|_|_ _ x
%e A262259 .       3             10
%e A262259 .
%e A262259 The symmetric representation of sigma(3) = 2 + 2 = 4 has two parts and they meet at the point (2, 2), so a(1) = 3.
%e A262259 The symmetric representation of sigma(10) = 9 + 9 = 18 has two parts and they meet at the point (7, 7), so a(2) = 10.
%e A262259 (End)
%e A262259 Also 10 is in the sequence because the concatenation of the widths of the symmetric representation of sigma(10) is 1111111110111111111 and it is a cyclops number (A134808). - _Omar E. Pol_, Aug 29 2021
%t A262259 (* test for membership in A174973 *)
%t A262259 a174973Q[n_]:=Module[{d=Divisors[n]}, Select[Rest[d] - 2 Most[d], #>0&]=={}]
%t A262259 a174973[n_]:=Select[Range[n], a174973Q]
%t A262259 (* compute numbers in the sequence *)
%t A262259 a262259[n_]:=Map[#(2#+1)&, Select[a174973[1, n], PrimeQ[2#+1]&]]
%t A262259 a262259[308] (* data *)
%Y A262259 Cf. A000203, A000217, A014105, A071561, A134808, A174973, A191363, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239929, A245092, A249351 (widths), A262045, A262048, A262626.
%K A262259 nonn,more
%O A262259 1,1
%A A262259 _Hartmut F. W. Hoft_, Sep 16 2015

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