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
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
KEYWORD
nonn
AUTHOR
Hartmut F. W. Hoft, Sep 22 2014
STATUS
approved