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.

1, 6, 3, 18, -1, 9, 30, 78, 15, 21, 162, -1, -1, -1, 81, 90, 666, 45, 75, 63, 147, 1458, -1, -1, -1, -1, -1, 729, 210, 1830, 135, 105, 165, 189, 357, 903, 450, -1, 225, -1, 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

Offset

1,2

Comments

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.

The single value T(10, 4) = -1 has been verified; see the conjecture below.

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.

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.

T(i, 1) <= 2 * 3^(i-1) and it is even for all i >1. Equality holds for all primes i.

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.

The positive numbers in columns 1..6 are subsequences of A174973, A239929, A279102, A280107, A320066, A320511, respectively.

Conjectures:

All entries T(i, j) in columns j >= 3 are odd.

T(i, 1)/2 is odd for all i > 1.

T(i, 1) = 2 * T(i, 3) for all nonprime i > 3, for i = 3, but not for i = 8.

T(i, 2)/2 is odd for all even i > 2.

T(i, 3) = A038547(i) for all nonprime i > 3, except i = 8.

T(2*i, 2*j) = -1 for j >= 2 and all prime i satisfying i >= prime(j+1).

From Omar E. Pol, Jun 08 2025: (Start)

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.

Observations:

At least for i < 12 if i is prime then T(i,1) = 2*T(i,i).

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)

Links

Table of n, a(n) for n=1..66.

Example

The first 12 rows of triangle T(i, j):
   i\j      1     2   3   4    5    6    7    8    9   10    11    12
   1:       1
   2:       6     3
   3:      18    -1   9
   4:      30    78  15  21
   5:     162    -1  -1  -1   81
   6:      90   666  45  75   63  147
   7:    1458    -1  -1  -1   -1   -1  729
   8:     210  1830 135 105  165  189  357  903
   9:     450    -1  25  -1 1225   -1  441   -1 3025
  10:     810 53622 405  -1 1377 1875  567 1539 4779 6875
  11:  118098    -1  -1  -1   -1   -1   -1   -1   -1   -1 59049
  12:     630 16290 315 495  525 1071 1287 1197 2499 6069 13915 29095
  ...

Mathematica

(* function partsSRS[ ] is defined in A377654 *)
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]
(* 0: not computed, value unknown. -1: no finite number exists. *)
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]
(* d rows that may contain 0s, computation through n *)
a384704[d_, n_] := First[SplitBy[Flatten[triangle[d, n]], #!=0&]]
a384704[11, 120000]

Crossrefs

Cf. A003056, A038547, A174973, A235791, A237048, A237591, A237593, A239929, A249223, A279102, A279387, A280107, A318843, A320066, A320511, A377654.

Sequence in context: A231881 A229005 A067990 * A174012 A050008 A166450

Adjacent sequences: A384701 A384702 A384703 * A384705 A384706 A384707

Keyword

sign,tabl

Author

Hartmut F. W. Hoft, Jun 07 2025

Status

approved