A386608 a(n) is the smallest number k for which the smallest part of the symmetric representation of sigma of k, SRS(k), has size n. If no such number exists, a(n) = -1.

1, 3, 2, 7, 21, 11, 4, 15, 10, 19, 57, 6, 69, 115, 8, 31, 93, 22, 37, 91, 26, 43, 45, 47, 141, 153, 34, 12, 265, 38, 16, 183, 242, 67, 201, 46, 73, 219, 18, 79, 237, 20, 105, 351, 58, 267, 273, 62, 52, 291, 101, 103, 309, 107, 109, 28, 74, 339, 565, 24, 1017, 861, 32, 127, 381, 86, 393, 135, 137, 76

Offset

1,2

Comments

Conjecture: Every a(n) is positive.

If even number n = p+q with p, q primes and 2 < p < q < 2p then SRS(p*q) = {(p*q+1)/2, p+q, (p*q+1)/2} so that 1 < a(p+q) <= p*q.

A351903(n) <= a(n), for all n >= 1 if the conjecture is true.

A351903(5) = 9 with SRS(9) = {5, 3, 5} and a(5) = 21 with SRS(21) = {11, 5, 5, 11}; thus 9 cannot be a member of this sequence.

a(54) = 107 with SRS(107) = {54, 54} and A351903(54) = 70 with SRS(70) = {54, 36, 54}; thus 107 cannot be a member of sequence A351903.

Conjecture: For each n >= 1, max( k : n = min(SRS(k)) ) <= n^2, with equality holding for odd primes and their powers (See the irregular table below; true for n <= 100).

Links

Table of n, a(n) for n=1..70.

Formula

a(n) = min( k : min(SRS(k)) = n ), and if no such k exists, a(n) = -1.

Example

a(2) = 3 since SRS(3) = {2, 2}, a(3) = 2 since SRS(2) = {3}, a(4) = 7 since SRS(7) = {4, 4};
a(5) = 21 since SRS(21) = {11, 5, 5, 11} and because in SRS(9) = {5, 3, 5} its central part 3 is the smallest.
Table of a(n), n <= 1.5 * 10^4, with column c containing those a(n) for which SRS(a(n)) consists of c parts. The numbers in each column are listed in the order of their positions in this sequence.
c: 1   2   3     4     5     6     7      8      9     ...
----------------------------------------------------------
   1   3   15    21    153   351   861    13065  66381
   2   7   91    57    273   1017  12369  27075  55917
   4   11  45    69    1225  1251  17019  27285  32319
   6   10  242   115   819   1791  17385  10017  74415
   8   19  135   93    825   1899  19647  35049  112413
   12  31  266   141   1911  2169  23115  24225  96393
   16  22  225   265   2079  1275  20313  45759  107085
   18  37  1156  183   3003  2547  17655  48531  135459
   20  26  315   201   3213  2637  22311  52017  80199
   28  43  748   219   3525  2853  37185  53697  84987
   ...
Except for 1, every number in column 1 is even since for odd k > 1, SRS(k) has at least 2 parts.
In column 2: 11 comes before 10 since a(6) = 11 with SRS(11) = {6, 6} and a(9) = 10 with SRS(10) = {9, 9}.
1310, 14210, 74450 are the smallest even numbers in columns 4, 5, and 6, respectively.
Up to n = 10^4 there are no even numbers in columns 7, 8 and 9.
Irregular table of sets { k : min(SRS(k)) = n } for n <= 27. The first column in the table is this sequence:
  1  | 1
  2  | 3
  3  | 2   5   9
  4  | 7
  5  | 21  25
  6  | 11  27
  7  | 4   13  33  49
  8  | 15  39  55
  9  | 10  17  65  81
  10 | 19  51
  11 | 57  85  121
  12 | 6   14  23  35  63  95  119
  13 | 69  133 169
  14 | 115 147 171
  15 | 8   29  50  125 161
  16 | 31  87  207
  17 | 93  145 253 289
  18 | 22  77  99  155 203 243 275
  19 | 37  217 261 361
  20 | 91  111 279 319
  21 | 26  41  98  117 185 341 377 441
  22 | 43  123 259 363 403
  23 | 45  129 205 333 465 529
  24 | 47  75  143 215 287 407
  25 | 141 301 369 481 625
  26 | 153 165 235 387 451 507 555 595
  27 | 34  53  329 473 533 629 729
  ...

Mathematica

(* function partsSRS[ ] is defined in A377654 *)
auvwxyz2[b_] := Module[{list=Table[0, b], found=0, k=1, min}, While[found<b, While[(min=Min[partsSRS[k]])>b||list[[min]]!=0, k++]; list[[min]]=k; found++]; list]
auvwxyz2[70] (* if the conjecture is true this function will always terminate *)

Crossrefs

Cf. A003056, A235791, A237048, A237591, A237593, A249223, A377654.

Sequence in context: A316087 A196537 A173099 * A111928 A348695 A248054

Adjacent sequences: A386605 A386606 A386607 * A386609 A386610 A386611

Keyword

nonn

Author

Hartmut F. W. Hoft, Oct 24 2025

Status

approved