A298855 Squarefree semiprimes p*q for which the symmetric representation of sigma(p*q) has four parts, in increasing order.

21, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 235, 237, 249, 253, 259, 265, 267, 287, 291, 295, 301, 303, 305, 309, 319, 321, 327, 329, 335, 339, 341, 355, 365, 371, 377, 381, 393, 395

Offset: 1

Author

Hartmut F. W. Hoft, Jan 27 2018

Keywords

• nonn
• tabf

Comments

All numbers in this sequence are odd since the symmetric representation of 2*p, p prime > 3, has two parts each of size 3*(p+1)/2, and that for 6 has one part of size 12.

A number in this sequence has the form p*q, p and q prime, 3 <= p and 2*p < q, since in this case 2*p <= floor((sqrt(8*p*q + 1) - 1)/2) < q so that 1's in row p*q of A237048 occur only in positions 1, 2, p and 2*p.

This sequence is a subsequence of A046388, hence of A006881, as well as of A174905, A241008 and A280107.

The two central parts of the symmetric representation of sigma(p*q), each of size (p+q)/2, meet on the diagonal when q = 2*p + 1 since in this case 2*p = floor((sqrt(8*p*q + 1) - 1)/2). These triangular numbers p*(2p+1) form sequence A156592, except for its first element 10, and form a subsequence of the diagonal in the associated irregular triangle of this sequence given in the Example section. They also are a subsequence of A264104. A function to compute the coordinates on the diagonal where the two central parts meet is defined in sequence A240542.

Except for missing 10 the intersection of this sequence and A298856 equals A156592.

Examples

			21=3*7 is the smallest number in the sequence since 2*3<7.
1081=23*(2*23+1) is in the sequence; its central parts meet at 751 on the diagonal.
The semiprimes p*q can be arranged as an irregular triangle with rows and columns labeled by the respective odd primes:
  q\p|   3    5    7   11   13   17   19   23
  ---+---------------------------------------
   7 |  21
  11 |  33   55
  13 |  39   65
  17 |  51   85  119
  19 |  57   95  133
  23 |  69  115  161  253
  29 |  87  145  203  319  377
  31 |  93  155  217  341  403
  37 | 111  185  259  407  481  629
  41 | 123  205  287  451  533  697  779
  43 | 129  215  301  473  559  731  817
  47 | 141  235  329  517  611  799  893 1081
		

Crossrefs

Cf. A001358, A005384, A005385, A006881, A046388, A068443, A156592, A174905, A237048, A237270, A237593, A240542, A241008, A264104, A280107, A298856.

Programs

Mathematica

(* Function a237270[] is defined in A237270 *)
a006881Q[n_] := Module[{f=FactorInteger[n]}, Length[f]==2 && AllTrue[Last[Transpose[f]], #==1&]]
a298855[m_, n_] := Select[Range[m, n], a006881Q[#] && Length[a237270[#]]==4 &]
a298855[1, 400] (* data *)
(* column for prime p through number n *)
stalk[n_, p_] := Select[a298855[1, n], First[First[FactorInteger[#]]]==p&]