A279102 Numbers n having three parts in the symmetric representation of sigma(n).

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

Offset: 1

Author

Hartmut F. W. Hoft, Dec 06 2016

Keywords

• nonn

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 n-th 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.

Examples

			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).
		

Crossrefs

Column 3 of A240062.

Cf. A087718, A174973 (column 1), A237048, A237270, A237271, A237593, A239929 (column 2), A249223, A251820, A262045, A279102.

Programs

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 *)