cp's OEIS Frontend

A247687 Numbers m with the property that the symmetric representation of sigma(m) has three parts of width one.

%I A247687 #24 Jun 27 2021 07:57:49
%S A247687 9,25,49,50,98,121,169,242,289,338,361,484,529,578,676,722,841,961,
%T A247687 1058,1156,1369,1444,1681,1682,1849,1922,2116,2209,2312,2738,2809,
%U A247687 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
%N A247687 Numbers m with the property that the symmetric representation of sigma(m) has three parts of width one.
%C A247687 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.
%C A247687 Since the sequence is defined by a two-parameter expression it can be written naturally as a triangle as shown in the Example section.
%C A247687 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
%H A247687 Hartmut F. W. Hoft, <a href="/A247687/a247687.pdf">Three regions width one - triangle formula proof</a>
%F A247687 As an irregular triangle, T(n, k) = 2^k * prime(n)^2 where n >= 2 and 0 <= k <= floor(log_2(prime(n)) - 1).
%e A247687 We show portions of the first eight columns, powers of two 0 <= k <= 7, and 55 rows of the triangle through prime(56) = 263.
%e A247687 p/k     0       1       2       3       4       5       6       7
%e A247687 3       9
%e A247687 5       25      50
%e A247687 7       49      98
%e A247687 11      121     242     484
%e A247687 13      169     338     676
%e A247687 17      289     578     1156    2312
%e A247687 19      361     722     1444    2888
%e A247687 23      529     1058    2116    4232
%e A247687 29      841     1682    3364    6728
%e A247687 31      961     1922    3844    7688
%e A247687 37      1369    2738    5476    10952   21904
%e A247687 41      1681    3362    6724    13448   26896
%e A247687 43      1849    3698    7396    14792   29584
%e A247687 47      2209    4418    8836    17672   35344
%e A247687 53      2809    5618    11236   22472   44944
%e A247687 59      3481    6962    13924   27848   55696
%e A247687 61      3721    7442    14884   29768   59536
%e A247687 67      4489    8978    17956   35912   71824   143648
%e A247687 71      5041    10082   20164   40328   80656   161312
%e A247687 .       .       .       .       .       .       .
%e A247687 .       .       .       .       .       .       .
%e A247687 131     17161   34322   68644   137288  274567  549152  1098304
%e A247687 137     18769   37538   75076   150152  300304  600608  1201216
%e A247687 .       .       .       .       .       .       .       .
%e A247687 .       .       .       .       .       .       .       .
%e A247687 257     66049   132098  264196  528392  1056784 2113568 4227136 8454272
%e A247687 263     69169   138338  276676  553352  1106704 2213408 4426816 8853632
%e A247687 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).
%t A247687 (* path[n] and a237270[n] are defined in A237270 *)
%t A247687 atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
%t A247687 (* data *)
%t A247687 Select[Range[10000], atmostOneDiagonalsQ[#] && Length[a237270[#]]==3 &]
%t A247687 (* expression for the triangle in the Example section *)
%t A247687 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]]]}]
%Y A247687 Cf. A000203, A237270, A237271, A237593, A241008, A241010, A246955, A250068, A250070, A250071.
%Y A247687 Cf. A070552, A263951.
%K A247687 nonn
%O A247687 1,1
%A A247687 _Hartmut F. W. Hoft_, Sep 22 2014