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

Author

Hartmut F. W. Hoft, Jun 07 2025

Keywords

• sign
• tabl

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)

Examples

			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
  ...
		

Crossrefs

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

Programs

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