A358862 a(n) is the smallest n-gonal number with exactly n distinct prime factors.

66, 44100, 11310, 103740, 3333330, 185040240, 15529888374, 626141842326, 21647593547580, 351877410344460, 82634328555218440, 2383985537862979050, 239213805711830629680

Offset

3,1

Comments

The corresponding indices of n-gonal numbers are 11, 210, 87, 228, 1155, 7854, 66612, 395646, 2193303, ...

Links

Table of n, a(n) for n=3..15.

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Eric Weisstein's World of Mathematics, Polygonal Number

Example

a(3) = 66, because 66 is a triangular number with 3 distinct prime factors {2, 3, 11} and this is the smallest such number.

Mathematica

Table[SelectFirst[PolygonalNumber[n, Range[400000]], PrimeNu[#]==n&], {n, 3, 10}] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Sep 09 2023 *)

Prog

(PARI) a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(n*k - n - 2*k + 4))\2); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 04 2022
(PARI)
omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A && ispolygonal(v, k), listput(list, v)), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(n < 3, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 04 2022

Crossrefs

Cf. A001221, A076551, A156236, A156237, A156238, A156239, A358863, A358864, A358865.

Sequence in context: A188453 A278848 A110150 * A295790 A246241 A337893

Adjacent sequences: A358859 A358860 A358861 * A358863 A358864 A358865

Keyword

nonn,more

Author

Ilya Gutkovskiy, Dec 03 2022

Extensions

a(12)-a(15) from Daniel Suteu, Dec 04 2022

Status

approved