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%I #23 Feb 12 2018 02:39:36
%S 21,55,253,406,1081,1378,1711,3403,3916,5671,9316,11026,13861,14878,
%T 15931,25651,27028,34453,36046,42778,50086,60031,64261,73153,75466,
%U 108811,114481,126253,129286,154846,158203,161596,171991,175528,212878,258121,298378,317206,326836,351541,366796,371953,392941
%N Numbers n with the property that the symmetric representation of sigma(n) has four parts, each of width one and two regions meet at the center of the Dyck path.
%C This sequence is a subsequence of A264102 and also of A014105, the second hexagonal numbers. Every number in this sequence is a triangular number.
%C The sequence A156592 of products of a Sophie Germain prime (A005384) and its associated safe prime (A005385) except for the first pair (2, 5) forms a subsequence of this sequence, the first column in the irregular triangular grid in the example.
%C The areas of the first two regions are (2^(m+1) - 1) * (2^(m+1) * p^2 * p + 1) / 2 and (2^(m+1) - 1) * (2^(m+1) * p + p + 1) / 2, respectively. Twice their sum equals sigma(n) = (2^(m+1) - 1) * (p + 1) * (2^(m+1) * p + 2).
%C For a proof of the formula for this sequence see the link.
%H Hartmut F. W. Hoft, <a href="/A264104/a264104.pdf">Diagram of symmetric representations of sigma(n), for n = 21, 55, 253, 406</a>
%H Hartmut F. W. Hoft, <a href="/A264104/a264104_1.pdf">Proof of 4 regions width 1 and 2 meet at center</a>
%F n = 2^m * p * (2^(m+1) * p + 1) where m >= 0, 2^(m+1) < p and p as well as 2^(m+1) * p + 1 are prime.
%e 406 = 2*7*29 is in the sequence since m = 1 and 4 < 7 < 28 < 29. The first two regions in the symmetric representation of sigma(406) = 720 start with legs 1 and 7 and have areas 306 and 54, respectively. Note also that 406 is a triangular number and the middle two regions meet at the center of the Dyck path.
%e 10 does not belong to this sequence since the symmetric representation of sigma(10) has two regions of width 1 that meet at the diagonal.
%e There is a natural arrangement of the numbers n = 2^m * p * (2^(m+1) * p + 1) as a sparse irregular triangular (p,m)-grid.
%e p\m| 0 1 2 3 4 5 ...
%e -------------------------------------------------------
%e 3 | 21
%e 5 | 55
%e 7 | 406
%e 11 | 253 3916
%e 13 | 1378
%e 17 | 9316
%e 19 |
%e 23 | 1081
%e 29 | 1711 27028
%e 31 |
%e 37 | 11026 175528
%e 41 | 3403
%e 43 | 14878
%e 47 |
%e 53 | 5671 1439056
%e 59 | 1783216
%e 61 | 476776
%e 67 | 36046 9195616
%e 71 | 161596 2582128
%e 73 | 42778 10916128
%e ...
%e The first number in the m = 6 column is 181880128 = 2^6*149*19073 in row p = 149 and the second is 228477376 = 2^6*167*21377 in row p = 167.
%t mStalk[m_, bound_] := Module[{p=NextPrime[2^(m+1)], list={}}, While[2^m*p*(2^(m+1)*p+1)<=bound, If[PrimeQ[2^(m+1)*p+1], AppendTo[list, 2^m *p*(2^(m+1)*p+1)]]; p=NextPrime[p]]; list]
%t a264104[bound_] := Module[{m=0, list={}}, While[2^m*NextPrime[2^(m+1)]*(2^(m+1)*NextPrime[2^(m+1)]+1)<=bound, list=Union[list, mStalk[m, bound]]; m++]; list]
%t a264104[400000] (* data *)
%Y Cf. A005384, A005385, A014105, A156592, A264102.
%Y For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.
%K nonn,tabf
%O 1,1
%A _Hartmut F. W. Hoft_, Nov 03 2015
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