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%I #33 Dec 16 2022 09:00:04
%S 1,4,10,460,9010,772210,20120860,1553569960,85507715710,
%T 14932196985010,1033664429333260,197628216951078460,
%U 21266854897681220860,7423007155473283614010,3108276166302017120182510,851452464506763307285599610,32749388246772812069108696710
%N a(n) is the smallest centered triangular number with exactly n distinct prime factors.
%C a(n) cannot be divisible by a bunch of primes like 3, 7, 11, 13, ... as (3*k^2 + 3*k + 2)/2 is never a multiple of any of them. - _David A. Corneth_, Dec 12 2022
%C a(16) <= 1421044357661885128003268103460. - _David A. Corneth_, Dec 14 2022
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredTriangularNumber.html">Centered Triangular Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%e a(4) = 9010, because 9010 is a centered triangular number with 4 distinct prime factors {2, 5, 17, 53} and this is the smallest such number.
%t c[k_] := (3*k^2 + 3*k + 2)/2; a[n_] := Module[{k = 0, ck}, While[PrimeNu[ck = c[k]] != n, k++]; ck]; Array[a, 9, 0] (* _Amiram Eldar_, Dec 09 2022 *)
%o (PARI) a(n) = for(k=0, oo, my(t=3*k*(k+1)/2 + 1); if(omega(t) == n, return(t))); \\ _Daniel Suteu_, Dec 10 2022
%Y Cf. A001221, A005448, A076551, A156329, A358894, A358929.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 06 2022
%E a(9)-a(11) from _Daniel Suteu_, Dec 10 2022
%E a(12)-a(13) from _David A. Corneth_, Dec 12 2022
%E a(13) corrected by _Daniel Suteu_, Dec 13 2022
%E a(14)-a(15) from _David A. Corneth_, Dec 14 2022
%E a(16) from _Daniel Suteu_, Dec 14 2022
%E a(15) corrected by _Daniel Suteu_, Dec 15 2022
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