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%I #41 Jun 11 2023 14:22:27
%S 0,2,1,13,19,7,61,331,169,1141,6487,898,20581,315826,59947,296143,
%T 1890466,6141994,1359025,49188715,20490901,264422320,178328878,
%U 1340590345,9476420614,5989636213,72238539832,103619599441,668478672403,794002910839,417430195531
%N a(n) is the least number k such that 1 + 2*k + 3*k^2 has exactly n prime divisors, counted with multiplicity.
%C a(n) is the least k such that A001222(A056109(k)) = n.
%H Gerry Martens, <a href="/A358205/b358205.txt">Table of n, a(n) for n = 0..40</a>
%e a(5) = 7 because 1 + 2*7 + 3*7^2 = 162 = 2*3^4 has 5 prime divisors, counted with multiplicity.
%e From _Jon E. Schoenfield_, Nov 05 2022: (Start)
%e 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:
%e .
%e n k = a(n) m = 1 + 2*k + 3*k^2
%e -- ------------ -----------------------------------------------------
%e 0 0 1
%e 1 2 17 (prime)
%e 2 1 6 = 2 * 3
%e 3 13 534 = 2 * 3 * 89
%e 4 19 1122 = 2 * 3 * 11 * 17
%e 5 7 162 = 2 * 3^4
%e 6 61 11286 = 2 * 3^3 * 11 * 19
%e 7 331 329346 = 2 * 3^4 * 19 * 107
%e 8 169 86022 = 2 * 3^6 * 59
%e 9 1141 3907926 = 2 * 3^5 * 11 * 17 * 43
%e 10 6487 126256482 = 2 * 3^5 * 11^2 * 19 * 113
%e 11 898 2421009 = 3^10 * 41
%e 12 20581 1270773846 = 2 * 3^9 * 19 * 1699
%e 13 315826 299238818481 = 3^9 * 19 * 73 * ...
%e 14 59947 10781048322 = 2 * 3^10 * 11 * 43 * 193
%e 15 296143 263102621634 = 2 * 3^12 * 17 * 14561
%e 16 1890466 10721588872401 = 3^12 * 11 * 19 * ...
%e 17 6141994 113172283172097 = 3^16 * 2629057
%e 18 1359025 5540849569926 = 2 * 3^14 * 11^2 * 4787
%e 19 49188715 7258589148431106 = 2 * 3^17 * 28103531
%e 20 20490901 1259631112357206 = 2 * 3^15 * 17 * 73 * ...
%e 21 264422320 209757490471391841 = 3^16 * 11 * 17 * ...
%e 22 178328878 95403566542874409 = 3^19 * 19 * 83 * ...
%e 23 1340590345 5391547422002837766 = 2 * 3^19 * 11^2 * ...
%e 24 9476420614 269407642979285252217 = 3^22 * 2617 * ...
%e 25 5989636213 107627225904222216534 = 2 * 3^20 * 19 * 97 * ...
%e 26 72238539832 15655219911322828844337 = 3^22 * 11 * 19 * ...
%e 27 103619599441 32211064165147101736326 = 2 * 3^22 * 11 * 43 * ...
%e 28 668478672403 1340591206374369138728034 = 2 * 3^22 * 19 * 331 * ...
%e 29 794002910839 1891321867264002956873442 = 2 * 3^23 * 11 * 73 * ...
%e 30 417430195531 522743904423981537506946 = 2 * 3^25 * 11 * 17 * ...
%e .
%e As a result, the last digits of the ternary representation of a(n) tend to fall into a pattern:
%e .
%e n a(n) a(n) in base 3
%e -- ------------ ---------------------------
%e 0 0 0_3
%e 1 2 2_3
%e 2 1 1_3
%e 3 13 111_3
%e 4 19 201_3
%e 5 7 21_3
%e 6 61 2021_3
%e 7 331 110021_3
%e 8 169 20021_3
%e 9 1141 1120021_3
%e 10 6487 22220021_3
%e 11 898 1020021_3
%e 12 20581 1001020021_3
%e 13 315826 121001020021_3
%e 14 59947 10001020021_3
%e 15 296143 120001020021_3
%e 16 1890466 10120001020021_3
%e 17 6141994 102120001020021_3
%e 18 1359025 2120001020021_3
%e 19 49188715 10102120001020021_3
%e 20 20490901 1102120001020021_3
%e 21 264422320 200102120001020021_3
%e 22 178328878 110102120001020021_3
%e 23 1340590345 10110102120001020021_3
%e 24 9476420614 220110102120001020021_3
%e 25 5989636213 120110102120001020021_3
%e 26 72238539832 20220110102120001020021_3
%e 27 103619599441 100220110102120001020021_3
%e 28 668478672403 2100220110102120001020021_3
%e 29 794002910839 2210220110102120001020021_3
%e 30 417430195531 1110220110102120001020021_3
%e (End)
%p N:= 18: # for a(0)..a(N)
%p V:= Array(0..N): count:= 0:
%p for k from 0 while count < N+1 do
%p v:= numtheory:-bigomega(1+2*k+3*k^2);
%p if v <= N and V[v] = 0 then
%p count:= count+1; V[v]:= k
%p fi
%p od:
%p convert(V,list);
%t a[n_] := Module[{i = 0},While[! PrimeOmega[1 + 2 i + 3 i^2] == n, i += 1]; i]
%t Table[a[n], {n, 0, 14}] (* _Gerry Martens_, Nov 05 2022 *)
%Y Cf. A001222, A056109, A086285, A122488.
%K nonn
%O 0,2
%A _Robert Israel_, Nov 03 2022
%E a(21)-a(22) from _Amiram Eldar_, Nov 04 2022
%E a(23)-a(30) from _Jon E. Schoenfield_, Nov 05 2022
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