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 A384704 #24 Jun 18 2025 21:56:42
%S A384704 1,6,3,18,-1,9,30,78,15,21,162,-1,-1,-1,81,90,666,45,75,63,147,1458,
%T A384704 -1,-1,-1,-1,-1,729,210,1830,135,105,165,189,357,903,450,-1,225,-1,
%U A384704 1225,-1,441,-1,3025,810,53622,405,-1,1377,1875,567,1539,4779,6875,118098,-1,-1,-1,-1,-1,-1,-1,-1,-1,59049
%N A384704 Triangle T(i, j), 1 <= j <= i, read by rows. T(i, j) is the smallest number k that has i odd divisors and whose symmetric representation of sigma, SRS(k), has j parts; when no such k exists then T(i, j) = -1.
%C A384704 T(i, j) = -1 for i >= 1 odd, nonprime, j even with 1 < j < i; also for i prime and all j with 1 < j < i.
%C A384704 The single value T(10, 4) = -1 has been verified; see the conjecture below.
%C A384704 T(i, i) <= 3^(i-1) for all i >=1 . Equality holds for all primes i. T(i, i) = A318843(i), for all i >= 1.
%C A384704 A038547(i) is the smallest number with exactly i odd divisors. Thus odd number A038547(i) occurs in row i of triangle T(i, j) so that A038547 is a subsequence of this sequence. For i prime, A038547(i) = T(i, i). For 4 <= i <= 10^9 nonprime, A038547(i) is in the third column, T(i, 3), except for i=8; furthermore, the first part of SRS(A038547(i)) has width 1 and size (A038547(i)+1)/2.
%C A384704 T(i, 1) <= 2 * 3^(i-1) and it is even for all i >1. Equality holds for all primes i.
%C A384704 T(i, 2) <= 2 * 3^(i/2-1) * p for all even i where p is the smallest prime greater than 4 * 3^(i/2-1). Equality holds when i = 2 * h where h is prime.
%C A384704 The positive numbers in columns 1..6 are subsequences of A174973, A239929, A279102, A280107, A320066, A320511, respectively.
%C A384704 Conjectures:
%C A384704 All entries T(i, j) in columns j >= 3 are odd.
%C A384704 T(i, 1)/2 is odd for all i > 1.
%C A384704 T(i, 1) = 2 * T(i, 3) for all nonprime i > 3, for i = 3, but not for i = 8.
%C A384704 T(i, 2)/2 is odd for all even i > 2.
%C A384704 T(i, 3) = A038547(i) for all nonprime i > 3, except i = 8.
%C A384704 T(2*i, 2*j) = -1 for j >= 2 and all prime i satisfying i >= prime(j+1).
%C A384704 From _Omar E. Pol_, Jun 08 2025: (Start)
%C A384704 T(i,j) is also the smallest number k whose symmetric representation of sigma(k) has i subparts and j parts, or -1 if no such k exists.
%C A384704 Observations:
%C A384704 At least for i < 12 if i is prime then T(i,1) = 2*T(i,i).
%C A384704 At least for i < 12 if i is prime then all terms in row i are -1's except the first and the last term. (End)
%e A384704 The first 12 rows of triangle T(i, j):
%e A384704 i\j 1 2 3 4 5 6 7 8 9 10 11 12
%e A384704 1: 1
%e A384704 2: 6 3
%e A384704 3: 18 -1 9
%e A384704 4: 30 78 15 21
%e A384704 5: 162 -1 -1 -1 81
%e A384704 6: 90 666 45 75 63 147
%e A384704 7: 1458 -1 -1 -1 -1 -1 729
%e A384704 8: 210 1830 135 105 165 189 357 903
%e A384704 9: 450 -1 25 -1 1225 -1 441 -1 3025
%e A384704 10: 810 53622 405 -1 1377 1875 567 1539 4779 6875
%e A384704 11: 118098 -1 -1 -1 -1 -1 -1 -1 -1 -1 59049
%e A384704 12: 630 16290 315 495 525 1071 1287 1197 2499 6069 13915 29095
%e A384704 ...
%t A384704 (* function partsSRS[ ] is defined in A377654 *)
%t A384704 setupT[d_] := Module[{list=Table[0, {i, d}, {j, i}], s, t}, For[s=1, s<=d, s++, For[t=1, t<=s, t++, If[(OddQ[s]&&Not[PrimeQ[s]]&&EvenQ[t]&&1<t<s)||(PrimeQ[s]&&1<t<s)||(s==10&&t==4), list[[s, t]]=-1]]]; list]
%t A384704 (* 0: not computed, value unknown. -1: no finite number exists. *)
%t A384704 triangle[d_, n_] := Module[{list=setupT[d], dL, pL, k=1}, While[k<=n, dL=Length[Select[Divisors[k], OddQ]]; pL=Length[partsSRS[k]]; If[pL<=dL<=d&&list[[dL, pL]]==0, list[[dL, pL]]=k]; k++]; list]
%t A384704 (* d rows that may contain 0s, computation through n *)
%t A384704 a384704[d_, n_] := First[SplitBy[Flatten[triangle[d, n]], #!=0&]]
%t A384704 a384704[11, 120000]
%Y A384704 Cf. A003056, A038547, A174973, A235791, A237048, A237591, A237593, A239929, A249223, A279102, A279387, A280107, A318843, A320066, A320511, A377654.
%K A384704 sign,tabl
%O A384704 1,2
%A A384704 _Hartmut F. W. Hoft_, Jun 07 2025