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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 12 20:12 EST 2019. Contains 329961 sequences. (Running on oeis4.)