A264104 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.

21, 55, 253, 406, 1081, 1378, 1711, 3403, 3916, 5671, 9316, 11026, 13861, 14878, 15931, 25651, 27028, 34453, 36046, 42778, 50086, 60031, 64261, 73153, 75466, 108811, 114481, 126253, 129286, 154846, 158203, 161596, 171991, 175528, 212878, 258121, 298378, 317206, 326836, 351541, 366796, 371953, 392941

Offset: 1

Author

Hartmut F. W. Hoft, Nov 03 2015

Keywords

• nonn
• tabf

Comments

This sequence is a subsequence of A264102 and also of A014105, the second hexagonal numbers. Every number in this sequence is a triangular number.

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.

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).

For a proof of the formula for this sequence see the link.

Examples

			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.
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.
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.
p\m| 0      1       2        3        4        5   ...
-------------------------------------------------------
3  | 21
5  | 55
7  |        406
11 | 253            3916
13 |        1378
17 |                9316
19 |
23 | 1081
29 | 1711           27028
31 |
37 |        11026           175528
41 | 3403
43 |        14878
47 |
53 | 5671                           1439056
59 |                                1783216
61 |                        476776
67 |        36046                            9195616
71 |                161596          2582128
73 |        42778                            10916128
...
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.
		

Links

• Hartmut F. W. Hoft, Diagram of symmetric representations of sigma(n), for n = 21, 55, 253, 406
• Hartmut F. W. Hoft, Proof of 4 regions width 1 and 2 meet at center

Crossrefs

Cf. A005384, A005385, A014105, A156592, A264102.

For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.

Programs

Mathematica

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]
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]
a264104[400000] (* data *)

Formula

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.