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%I #13 Jan 28 2018 13:14:50
%S 21,33,39,51,55,57,65,69,85,87,93,95,111,115,119,123,129,133,141,145,
%T 155,159,161,177,183,185,201,203,205,213,215,217,219,235,237,249,253,
%U 259,265,267,287,291,295,301,303,305,309,319,321,327,329,335,339,341,355,365,371,377,381,393,395
%N Squarefree semiprimes p*q for which the symmetric representation of sigma(p*q) has four parts, in increasing order.
%C 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.
%C 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.
%C This sequence is a subsequence of A046388, hence of A006881, as well as of A174905, A241008 and A280107.
%C 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.
%C Except for missing 10 the intersection of this sequence and A298856 equals A156592.
%e 21=3*7 is the smallest number in the sequence since 2*3<7.
%e 1081=23*(2*23+1) is in the sequence; its central parts meet at 751 on the diagonal.
%e The semiprimes p*q can be arranged as an irregular triangle with rows and columns labeled by the respective odd primes:
%e q\p| 3 5 7 11 13 17 19 23
%e ---+---------------------------------------
%e 7 | 21
%e 11 | 33 55
%e 13 | 39 65
%e 17 | 51 85 119
%e 19 | 57 95 133
%e 23 | 69 115 161 253
%e 29 | 87 145 203 319 377
%e 31 | 93 155 217 341 403
%e 37 | 111 185 259 407 481 629
%e 41 | 123 205 287 451 533 697 779
%e 43 | 129 215 301 473 559 731 817
%e 47 | 141 235 329 517 611 799 893 1081
%t (* Function a237270[] is defined in A237270 *)
%t a006881Q[n_] := Module[{f=FactorInteger[n]}, Length[f]==2 && AllTrue[Last[Transpose[f]], #==1&]]
%t a298855[m_, n_] := Select[Range[m, n], a006881Q[#] && Length[a237270[#]]==4 &]
%t a298855[1, 400] (* data *)
%t (* column for prime p through number n *)
%t stalk[n_, p_] := Select[a298855[1, n], First[First[FactorInteger[#]]]==p&]
%Y Cf. A001358, A005384, A005385, A006881, A046388, A068443, A156592, A174905, A237048, A237270, A237593, A240542, A241008, A264104, A280107, A298856.
%K nonn,tabf
%O 1,1
%A _Hartmut F. W. Hoft_, Jan 27 2018
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