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A247687 Numbers n with the property that the symmetric representation of sigma(n) has three parts of width one. 10
9, 25, 49, 50, 98, 121, 169, 242, 289, 338, 361, 484, 529, 578, 676, 722, 841, 961, 1058, 1156, 1369, 1444, 1681, 1682, 1849, 1922, 2116, 2209, 2312, 2738, 2809, 2888, 3362, 3364, 3481, 3698, 3721, 3844, 4232, 4418, 4489, 5041, 5329, 5476, 5618, 6241, 6724, 6728, 6889, 6962, 7396, 7442, 7688, 7921, 8836, 8978, 9409 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The symmetric representation of sigma(n) has 3 regions of width 1 where the two extremal regions each have 2^k - 1 legs and the central region starts with the p-th leg of the associated Dyck path for sigma(n) precisely when n = 2^(k - 1) * p^2 where 2^k < p <= row(n), k >= 1, p >= 3 is prime and row(n) = floor((sqrt(8*n + 1) - 1)/2). Furthermore, the areas of the two outer regions are (2^k - 1)*(p^2 + 1)/2 each so that the area of the central region is (2^k - 1)*p - for a proof see the link.

Since the sequence a(n, k) is defined by a two-parameter expression it can be written naturally as a triangle as shown in the Example section.

LINKS

Table of n, a(n) for n=1..57.

Hartmut F. W. Hoft, Three regions width one - triangle formula proof

FORMULA

Formula for the triangle of numbers associated with the sequence:

a(n, k) = 2^k * prime(n)^2 where n >= 2, 0 <= k <= floor(log2(prime(n)) - 1), and log2( ) is the logarithm to base 2.

EXAMPLE

We show portions of the first eight columns, powers of two 0 <= k <= 7, and 55 rows of the triangle through prime(56) = 263.

p/k     0       1       2       3       4       5       6       7

3       9

5       25      50

7       49      98

11      121     242     484

13      169     338     676

17      289     578     1156    2312

19      361     722     1444    2888

23      529     1058    2116    4232

29      841     1682    3364    6728

31      961     1922    3844    7688

37      1369    2738    5476    10952   21904

41      1681    3362    6724    13448   26896

43      1849    3698    7396    14792   29584

47      2209    4418    8836    17672   35344

53      2809    5618    11236   22472   44944

59      3481    6962    13924   27848   55696

61      3721    7442    14884   29768   59536

67      4489    8978    17956   35912   71824   143648

71      5041    10082   20164   40328   80656   161312

.       .       .       .       .       .       .

.       .       .       .       .       .       .

131     17161   34322   68644   137288  274567  549152  1098304

137     18769   37538   75076   150152  300304  600608  1201216

.       .       .       .       .       .       .       .

.       .       .       .       .       .       .       .

257     66049   132098  264196  528392  1056784 2113568 4227136 8454272

263     69169   138338  276676  553352  1106704 2213408 4426816 8853632

Number 4 is not in this sequence since the symmetric representation of sigma(4) consists of a single region. Column k=0 contains the squares of primes (A001248(n), n>=2), column k=1 contains double the squares of primes (A079704(n), n>=2) and column k=2 contains four times the squares of primes (A069262(n), n>=5).

MATHEMATICA

(* path[n] and a237270[n] are defined in A237270 *)

atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]

(* data *)

Select[Range[10000], atmostOneDiagonalsQ[#] && Length[a237270[#]]==3 &]

(* expression for the triangle in the Example section *)

TableForm[Table[2^k Prime[n]^2, {n, 2, 57}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth -> 2, TableHeadings -> {Map[Prime, Range[2, 57]], Range[0, Floor[Log[2, Prime[57] - 1]]]}]

CROSSREFS

Cf. A000203, A237270, A237271, A237593, A241008, A241010, A246955, A250068, A250070, A250071.

Sequence in context: A031036 A291259 A051132 * A075026 A113659 A325701

Adjacent sequences:  A247684 A247685 A247686 * A247688 A247689 A247690

KEYWORD

nonn

AUTHOR

Hartmut F. W. Hoft, Sep 22 2014

STATUS

approved

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Last modified May 21 11:12 EDT 2019. Contains 323443 sequences. (Running on oeis4.)