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%I #13 Oct 29 2025 18:43:15
%S 1,3,2,7,21,11,4,15,10,19,57,6,69,115,8,31,93,22,37,91,26,43,45,47,
%T 141,153,34,12,265,38,16,183,242,67,201,46,73,219,18,79,237,20,105,
%U 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
%N 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.
%C Conjecture: Every a(n) is positive.
%C 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.
%C A351903(n) <= a(n), for all n >= 1 if the conjecture is true.
%C 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.
%C 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.
%C 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).
%F a(n) = min( k : min(SRS(k)) = n ), and if no such k exists, a(n) = -1.
%e a(2) = 3 since SRS(3) = {2, 2}, a(3) = 2 since SRS(2) = {3}, a(4) = 7 since SRS(7) = {4, 4};
%e 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.
%e 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.
%e c: 1 2 3 4 5 6 7 8 9 ...
%e ----------------------------------------------------------
%e 1 3 15 21 153 351 861 13065 66381
%e 2 7 91 57 273 1017 12369 27075 55917
%e 4 11 45 69 1225 1251 17019 27285 32319
%e 6 10 242 115 819 1791 17385 10017 74415
%e 8 19 135 93 825 1899 19647 35049 112413
%e 12 31 266 141 1911 2169 23115 24225 96393
%e 16 22 225 265 2079 1275 20313 45759 107085
%e 18 37 1156 183 3003 2547 17655 48531 135459
%e 20 26 315 201 3213 2637 22311 52017 80199
%e 28 43 748 219 3525 2853 37185 53697 84987
%e ...
%e Except for 1, every number in column 1 is even since for odd k > 1, SRS(k) has at least 2 parts.
%e 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}.
%e 1310, 14210, 74450 are the smallest even numbers in columns 4, 5, and 6, respectively.
%e Up to n = 10^4 there are no even numbers in columns 7, 8 and 9.
%e Irregular table of sets { k : min(SRS(k)) = n } for n <= 27. The first column in the table is this sequence:
%e 1 | 1
%e 2 | 3
%e 3 | 2 5 9
%e 4 | 7
%e 5 | 21 25
%e 6 | 11 27
%e 7 | 4 13 33 49
%e 8 | 15 39 55
%e 9 | 10 17 65 81
%e 10 | 19 51
%e 11 | 57 85 121
%e 12 | 6 14 23 35 63 95 119
%e 13 | 69 133 169
%e 14 | 115 147 171
%e 15 | 8 29 50 125 161
%e 16 | 31 87 207
%e 17 | 93 145 253 289
%e 18 | 22 77 99 155 203 243 275
%e 19 | 37 217 261 361
%e 20 | 91 111 279 319
%e 21 | 26 41 98 117 185 341 377 441
%e 22 | 43 123 259 363 403
%e 23 | 45 129 205 333 465 529
%e 24 | 47 75 143 215 287 407
%e 25 | 141 301 369 481 625
%e 26 | 153 165 235 387 451 507 555 595
%e 27 | 34 53 329 473 533 629 729
%e ...
%t (* function partsSRS[ ] is defined in A377654 *)
%t 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]
%t auvwxyz2[70] (* if the conjecture is true this function will always terminate *)
%Y Cf. A003056, A235791, A237048, A237591, A237593, A249223, A377654.
%K nonn
%O 1,2
%A _Hartmut F. W. Hoft_, Oct 24 2025