

A279102


Numbers n having three parts in the symmetric representation of sigma(n).


6



9, 15, 25, 35, 45, 49, 50, 70, 77, 91, 98, 110, 121, 130, 135, 143, 154, 169, 170, 182, 187, 190, 209, 221, 225, 238, 242, 247, 266, 286, 289, 299, 315, 322, 323, 338, 350, 361, 374, 391, 405, 418, 437, 442, 484, 493, 494, 506, 527, 529, 550, 551, 572, 578, 589, 598, 638, 646, 650, 667, 675, 676, 682
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OFFSET

1,1


COMMENTS

Let n = 2^m * q with m >= 0 and q odd, let row(n) = floor(sqrt(8*n+1)  1)/2), and let 1 = d_1 < ... < d_h <= row(n) < d_(h+1) < ... < d_k = q be the k odd divisors of n.
The symmetric representation of sigma(n) consists of 3 parts precisely when there is a unique i, 1 <= i < h, such that 2^(m+1) * d_i < d_(i+1) and d_h <= row(n) < 2^(m+1) * d_h.
This property of the odd divisors of n is equivalent to the nth row of the irregular triangle of A249223 consisting of a block of positive numbers, followed by a block of zeros, followed in turn by a block of positive numbers, i.e., determining the first part and the left half of the center part of the symmetric representation of sigma(n), resulting in 3 parts.
Let n be the product of two primes p and q satisfying 2 < p < q < 2*p. Then n satisfies the property above so that the odd numbers in A087718 form a subsequence.


LINKS

Table of n, a(n) for n=1..63.


EXAMPLE

a(4) = 35 = 5*7 is in the sequence since 1 < 2 < 5 < row(35) = 7 < 10;
a(8) = 70 = 2*5*7 is in the sequence since 1 < 4 < 5 < row(70) = 11 < 20;
140 = 4*5*7 is not in the sequence since 1 < 5 < 7 < 8 < row(140) = 16 < 20;
a(506) = 5950 = 2*25*7*17 is in the sequence since 1*4 < 5 is the only pair of odd divisors 1 < 5 < 7 < 17 < 25 < 35 < 85 < row(5950) = 108 satisfying the property (see A251820).


MATHEMATICA

(* support functions are defined in A237048 and A262045 *)
segmentsSigma[n_] := Length[Select[SplitBy[a262045[n], #!=0&], First[#]!=0&]]
a279102[m_, n_] := Select[Range[m, n], segmentsSigma[#]==3&]
a279102[1, 700] (* sequence data *)
(* An equivalent, but slower computation is based on A237271 *)
a279102[m_, n_] := Select[Range[m, n], a237271[#]==3&]
a279102[1, 700] (* sequence data *)


CROSSREFS

Column 3 of A240062.
Cf. A087718, A174973 (column 1), A237048, A237270, A237271, A237593, A239929 (column 2), A249223, A251820, A262045, A279102.
Sequence in context: A079290 A176404 A227198 * A319529 A249730 A251415
Adjacent sequences: A279099 A279100 A279101 * A279103 A279104 A279105


KEYWORD

nonn


AUTHOR

Hartmut F. W. Hoft, Dec 06 2016


STATUS

approved



