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A247687 Numbers m with the property that the symmetric representation of sigma(m) has three parts of width one. 18
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(m) 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(m) precisely when m = 2^(k - 1) * p^2 where 2^k < p <= row(m), k >= 1, p >= 3 is prime and row(m) = floor((sqrt(8*m + 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 is defined by a two-parameter expression it can be written naturally as a triangle as shown in the Example section.

A263951 is a subsequence of this sequence containing the squares of all those primes p for which the areas of the 3 regions in the symmetric representation of p^2 (p once and (p^2 + 1)/2 twice) are primes; i.e., p^2 and p^2 + 1 are semiprimes (see A070552). - Hartmut F. W. Hoft, Aug 06 2020

LINKS

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

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

FORMULA

As an irregular triangle, T(n, k) = 2^k * prime(n)^2 where n >= 2 and 0 <= k <= floor(log_2(prime(n)) - 1).

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.

Cf. A070552, A263951.

Sequence in context: A348749 A291259 A051132 * A075026 A339727 A339128

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 January 29 17:21 EST 2023. Contains 359923 sequences. (Running on oeis4.)