A358928 a(n) is the smallest centered triangular number with exactly n distinct prime factors.

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

Table of n, a(n) for n=0..16.

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

Cf. A001221, A005448, A076551, A156329, A358894, A358929.

Sequence in context: A321795 A220197 A299880 * A085649 A152396 A326720

Adjacent sequences: A358925 A358926 A358927 * A358929 A358930 A358931

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