This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A338486 #10 Nov 23 2020 07:56:09
%S A338486 15,35,45,70,77,91,110,130,135,143,154,170,182,187,190,209,221,225,
%T A338486 238,247,266,286,299,322,323,350,374,391,405,418,437,442,493,494,506,
%U A338486 527,550,551,572,589,598,638,646,650,667,682,703,713,748,754,782,806,814,836,850
%N A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.
%C A338486 This sequence is a subsequence of A279102. The definition of the sequence excludes squares of primes, A001248, since the 3 regions of their symmetric representation of sigma have width 1 (first column in the irregular triangle of A247687).
%C A338486 Table of numbers in this sequence arranged by the number of prime factors, counting multiplicities:
%C A338486 2 3 4 5 6 7 ...
%C A338486 ------------------------------------------
%C A338486 15 45 135 405 1215 3645
%C A338486 35 70 225 1125 5625 ...
%C A338486 77 110 350 1750 8750 744795
%C A338486 91 130 550 2584 ... ...
%C A338486 143 154 572 2750 85455
%C A338486 187 170 650 3128 ...
%C A338486 209 182 748 3250
%C A338486 221 190 836 3496
%C A338486 247 238 850 3944
%C A338486 299 266 884 4216
%C A338486 ... ... ... ...
%C A338486 1035 9585
%C A338486 ... ...
%C A338486 The numbers in the first row of the table above are b(k) = 5*3^k, k>=1, (see A005030) so that infinitely many odd numbers occur outside of the first column. The central region of the symmetric representation of sigma(b(k)) contains 2*k-1 separate contiguous sections consisting of sequences of entire legs of width 2, k>=1 (see Lemma 2 in the link).
%C A338486 Conjecture: The combined extent of these sections in sigma(b(k)) is 2*3^(k-1) - 1 = A048473(k-1), k>=1.
%C A338486 Since each number n in the first column and first row has a prime factor of odd exponent a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 2. For odd numbers n not in the first row or column in which all prime factors have even powers, such as 225 and 5625 in the second row, a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 1 (see Lemma 1 in the link).
%C A338486 For each k>=3 and every prime p such that b(k-1) < 2*p < 4*b(k-2), the odd number p*b(k-1) is in the column of b(k). The two inequalities are equivalent to b(k-1) <= row(p*b(k-1)) < 2*b(k-1) ensuring that the symmetric representation of sigma(p*b(k-1)) consists of 3 regions.
%C A338486 45 is the only odd number in its column (see Lemma 3 in the link).
%C A338486 Since the factors of n = p*q satisfy 2 < p < q < 2*p the first column in the table above is a subsequence of A082663 and of A087718 (see Lemma 4 in the link). Each of the two outer regions consists of a single leg of width 1 and length (1 + p*q)/2. The center region of size p+q consists of two subparts (see A196020 & A280851) of width 1 of sizes 2*p-q and 2*q-p, respectively (see Lemma 5 in the link). The table below arranges the first column in the table above according to the length 2*p-q of their single contiguous extent of width 2 in the center region:
%C A338486 1 3 5 7 9 11 13 15 ...
%C A338486 ------------------------------------------------------
%C A338486 15 35 187 247 143 391 2257 323
%C A338486 91 77 493 589 221 1363 3139 437
%C A338486 703 209 943 2479 551 2911 6649 713
%C A338486 1891 299 1537 3397 851 3901 ... 1247
%C A338486 2701 527 4183 8509 1643 6313 1457
%C A338486 ... ... ... ... ... ... ....
%C A338486 A129521: p*q satisfies 2*p - q = 1 (excluding A129521(1)=6)
%C A338486 A226755: p*q satisfies 2*p - q = 3 (excluding A226755(1)=9)
%C A338486 Sequences with larger differences 2*p - q are not in OEIS.
%H A338486 Hartmut F. W. Hoft, <a href="/A338486/a338486.pdf">proofs for quoted lemmas</a>
%e A338486 a(6) = 91 = 7*13 is in the sequence and in the 2-column of the first table since 1 < 2 < 7 < 13 = row(91) representing the 4 odd divisors 1 - 91 - 7 - 13 (see A237048) results in the following pattern for the widths of the legs (see A249223): 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2. It also is in the 1-column of the second table since it has a single area of width 2 which is 1 unit long.
%e A338486 a(29) = 405 = 5*3^4 is in the sequence and in the 5-column of the first table since 1 < 2 < 3 < 5 < 6 < 9 < 10 < 15 < 18 < 27 = row(405) representing the 10 odd divisors 1 - 405 - 3 - 5 - 135 - 9 - 81 - 15 - 45 - 27 results in the following pattern for the widths of the legs: 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2, and 7 = 2*4 - 1 sections of width 2 in the central region.
%e A338486 a(35) = 506 = 2*11*23 is in the sequence since positions 1 < 4 < 11 < 23 < row(506) = 31 representing the 4 odd divisors 1 - 253 - 11 - 23 results in the following pattern for the widths of the legs: 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2 for 3 regions with width not exceeding 2, with the two outer regions consisting of 3 legs of width 1, and a single area of width 2 in the central region.
%t A338486 (* Functions path and a237270 are defined in A237270 *)
%t A338486 maxDiagonalLength[n_] := Max[Map[#[[1]]-#[[2]]&, Transpose[{Drop[Drop[path[n], 1], -1], path[n-1]}]]]
%t A338486 a338486[m_, n_] := Module[{r, list={}, k}, For[k=m, k<=n, k++, r=a237270[k]; If[Length[r]== 3 && maxDiagonalLength[k]==2,AppendTo[list, k]]]; list]
%t A338486 a338486[1, 850]
%Y A338486 Cf. A001248, A005030, A048473, A082663, A087718, A129521, A196020, A226755, A235791, A237048, A237270, A237271, A237591, A237593, A247687, A249223, A279102, A280107, A280851.
%K A338486 nonn
%O A338486 1,1
%A A338486 _Hartmut F. W. Hoft_, Oct 30 2020