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A358928
a(n) is the smallest centered triangular number with exactly n distinct prime factors.
2
1, 4, 10, 460, 9010, 772210, 20120860, 1553569960, 85507715710, 14932196985010, 1033664429333260, 197628216951078460, 21266854897681220860, 7423007155473283614010, 3108276166302017120182510, 851452464506763307285599610, 32749388246772812069108696710
OFFSET
0,2
COMMENTS
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
a(16) <= 1421044357661885128003268103460. - David A. Corneth, Dec 14 2022
LINKS
Eric Weisstein's World of Mathematics, Centered Triangular Number
Eric Weisstein's World of Mathematics, Distinct Prime Factors
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 06 2022
EXTENSIONS
a(9)-a(11) from Daniel Suteu, Dec 10 2022
a(12)-a(13) from David A. Corneth, Dec 12 2022
a(13) corrected by Daniel Suteu, Dec 13 2022
a(14)-a(15) from David A. Corneth, Dec 14 2022
a(16) from Daniel Suteu, Dec 14 2022
a(15) corrected by Daniel Suteu, Dec 15 2022
STATUS
approved