

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.


2



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
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OFFSET

1,1


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.


LINKS

Table of n, a(n) for n=1..43.
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


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.


EXAMPLE

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.


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 *)


CROSSREFS

Cf. A005384, A005385, A014105, A156592, A264102.
For symmetric representation of sigma: A235791, A236104, A237270, A237271, A237591, A237593, A241008, A246955.
Sequence in context: A180674 A067431 A083676 * A236694 A292368 A301607
Adjacent sequences: A264101 A264102 A264103 * A264105 A264106 A264107


KEYWORD

nonn,tabf


AUTHOR

Hartmut F. W. Hoft, Nov 03 2015


STATUS

approved



