%I #22 Dec 06 2016 22:16:38
%S 3,10,44,78,136,348,592,666,820,1272,1652,1830,2144,2628,3320,3738,
%T 4656,5886,6328,7620,8384,9042,10728,13040,14532,15752,16290,18528,
%U 21100,21944,24084,25424,28920,32382,32896,35508,39340,42192,46050,48828
%N a(n), n>1, is the smallest number k whose symmetric representation of sigma(k) has two parts and has a larger number of legs in its two parts than a(n-1); a(1)=3.
%C A number k with two parts in its symmetric representation of sigma(k) [ssrs(k) = 2] has the form k = q*p with q in A174973, p prime and 2*q < p. This implies that 2*q <= row(k) < p and the first 0 in the k-th row of A249223 (having row(k) = floor((sqrt(8*k+1)-1)/2) entries) occurs at position 2*q so that 2*q-1 is the number of legs in each of the two parts. Therefore, the numbers 2*q-1 with q in A174973 are the only possible leg counts when ssrs(k) = 2, and for given q in A174973 and smallest prime p(q) > 2*q the number k = q*p(q) is the smallest with a leg count of 2*q-1. Consequently, each number q*p in the column of the irregular triangle A239929 labeled by q in A174793 with p prime satisfies ssrs(q*p) = 2*q-1.
%C a(1) = 3 is the only odd number since 1 is the only odd number in A174973.
%C Every number n = 2^m * p, m >= 0, 2^(m+1) < p and p prime, in this sequence is the sum of 2^(m+1) consecutive positive integers which includes every number in A246956.
%F a(n) = A174973(n) * A007918(2 * A174973(n) + 1).
%e a(3)=44 is the smallest number whose symmetric representation has 2 parts and 7 legs in each part.
%e a(4)=78 is the smallest number whose symmetric representation has 2 parts and 11 legs in each part.
%e No number k whose symmetric representation of sigma(k) has 2 parts can have 21 legs in its parts since there is no q in A174973 such that 2*q - 1 = 21.
%t a174973Q[n_] := Module[{d=Divisors[n]}, Select[Rest[d] - 2*Most[d], #>0&]=={}]
%t a279105[n_] := Map[# * NextPrime[2*#]&, Select[Range[n], a174973Q]]
%t a279105[150] (* sequence data *)
%Y Right border of A239929.
%Y Supersequence of A246956 and A262259.
%Y Cf. A007918, A174973, A237048, A237270, A237271, A237593, A246956, A249223, A279105.
%K nonn
%O 1,1
%A _Hartmut F. W. Hoft_, Dec 06 2016