A358205 a(n) is the least number k such that 1 + 2*k + 3*k^2 has exactly n prime divisors, counted with multiplicity.

0, 2, 1, 13, 19, 7, 61, 331, 169, 1141, 6487, 898, 20581, 315826, 59947, 296143, 1890466, 6141994, 1359025, 49188715, 20490901, 264422320, 178328878, 1340590345, 9476420614, 5989636213, 72238539832, 103619599441, 668478672403, 794002910839, 417430195531

Offset

0,2

Comments

a(n) is the least k such that A001222(A056109(k)) = n.

Links

Gerry Martens, Table of n, a(n) for n = 0..40

Example

a(5) = 7 because 1 + 2*7 + 3*7^2 = 162 = 2*3^4 has 5 prime divisors, counted with multiplicity.
From Jon E. Schoenfield, Nov 05 2022: (Start)
Let m = 1 + 2*k + 3*k^2. Since no such number m is divisible by 2^2, 5, or 7, the smallest number m having a given number of prime factors counted with multiplicity will tend to have a large number of 3's among its prime factors:
.
   n    k = a(n)                     m = 1 + 2*k + 3*k^2
  --  ------------  -----------------------------------------------------
   0             0                          1
   1             2                         17 (prime)
   2             1                          6 = 2 * 3
   3            13                        534 = 2 * 3    * 89
   4            19                       1122 = 2 * 3    * 11 * 17
   5             7                        162 = 2 * 3^4
   6            61                      11286 = 2 * 3^3  * 11 * 19
   7           331                     329346 = 2 * 3^4  * 19 * 107
   8           169                      86022 = 2 * 3^6  * 59
   9          1141                    3907926 = 2 * 3^5  * 11 * 17 * 43
  10          6487                  126256482 = 2 * 3^5  * 11^2 * 19 * 113
  11           898                    2421009 =     3^10 * 41
  12         20581                 1270773846 = 2 * 3^9  * 19 * 1699
  13        315826               299238818481 =     3^9  * 19 * 73 * ...
  14         59947                10781048322 = 2 * 3^10 * 11 * 43 * 193
  15        296143               263102621634 = 2 * 3^12 * 17 * 14561
  16       1890466             10721588872401 =     3^12 * 11 * 19 * ...
  17       6141994            113172283172097 =     3^16 * 2629057
  18       1359025              5540849569926 = 2 * 3^14 * 11^2 * 4787
  19      49188715           7258589148431106 = 2 * 3^17 * 28103531
  20      20490901           1259631112357206 = 2 * 3^15 * 17 * 73 * ...
  21     264422320         209757490471391841 =     3^16 * 11 * 17 * ...
  22     178328878          95403566542874409 =     3^19 * 19 * 83 * ...
  23    1340590345        5391547422002837766 = 2 * 3^19 * 11^2 * ...
  24    9476420614      269407642979285252217 =     3^22 * 2617 * ...
  25    5989636213      107627225904222216534 = 2 * 3^20 * 19 * 97 * ...
  26   72238539832    15655219911322828844337 =     3^22 * 11 * 19 * ...
  27  103619599441    32211064165147101736326 = 2 * 3^22 * 11 * 43 * ...
  28  668478672403  1340591206374369138728034 = 2 * 3^22 * 19 * 331 * ...
  29  794002910839  1891321867264002956873442 = 2 * 3^23 * 11 * 73 * ...
  30  417430195531   522743904423981537506946 = 2 * 3^25 * 11 * 17 * ...
.
As a result, the last digits of the ternary representation of a(n) tend to fall into a pattern:
.
   n      a(n)             a(n) in base 3
  --  ------------  ---------------------------
   0             0                          0_3
   1             2                          2_3
   2             1                          1_3
   3            13                        111_3
   4            19                        201_3
   5             7                         21_3
   6            61                       2021_3
   7           331                     110021_3
   8           169                      20021_3
   9          1141                    1120021_3
  10          6487                   22220021_3
  11           898                    1020021_3
  12         20581                 1001020021_3
  13        315826               121001020021_3
  14         59947                10001020021_3
  15        296143               120001020021_3
  16       1890466             10120001020021_3
  17       6141994            102120001020021_3
  18       1359025              2120001020021_3
  19      49188715          10102120001020021_3
  20      20490901           1102120001020021_3
  21     264422320         200102120001020021_3
  22     178328878         110102120001020021_3
  23    1340590345       10110102120001020021_3
  24    9476420614      220110102120001020021_3
  25    5989636213      120110102120001020021_3
  26   72238539832    20220110102120001020021_3
  27  103619599441   100220110102120001020021_3
  28  668478672403  2100220110102120001020021_3
  29  794002910839  2210220110102120001020021_3
  30  417430195531  1110220110102120001020021_3
(End)

Maple

N:= 18: # for a(0)..a(N)
V:= Array(0..N): count:= 0:
for k from 0 while count < N+1 do
  v:= numtheory:-bigomega(1+2*k+3*k^2);
if v <= N and V[v] = 0 then
    count:= count+1; V[v]:= k
fi
od:
convert(V, list);

Mathematica

a[n_] := Module[{i = 0}, While[! PrimeOmega[1 + 2 i + 3 i^2] == n, i += 1]; i]
Table[a[n], {n, 0, 14}] (* Gerry Martens, Nov 05 2022 *)

Crossrefs

Cf. A001222, A056109, A086285, A122488.

Sequence in context: A074808 A113097 A032001 * A118679 A087451 A063558

Adjacent sequences: A358202 A358203 A358204 * A358206 A358207 A358208

Keyword

nonn

Author

Robert Israel, Nov 03 2022

Extensions

a(21)-a(22) from Amiram Eldar, Nov 04 2022

a(23)-a(30) from Jon E. Schoenfield, Nov 05 2022

Status

approved