OFFSET
2,1
COMMENTS
The corresponding n-gonal index is A359014(n).
a(n) is the first n-gonal number k such that A001222(k) = n. - Robert Israel, Jan 15 2023
LINKS
Eric Weisstein's World of Mathematics, Prime Factor.
Eric Weisstein's World of Mathematics, Polygonal Number.
Robert G. Wilson v, Table for n, index of polygonal number, a(n) and Factorization of a(n), for n=1..36.
FORMULA
A001222(a(n)) = n. - Robert Israel, Jan 15 2023
EXAMPLE
a(3) = 28, because 28 is a triangular number with 3 prime factors (counted with multiplicity) {2, 2, 7} and this is the smallest such number.
MAPLE
g:= proc(s) local n, p, F;
for n from 1 to 10^7 do
p:= (s-2)*n*(n-1)/2 + n;
if numtheory:-bigomega(p) = s then return p fi;
od
end proc:
map(g, [$2..30]); # Robert Israel, Jan 15 2023
MATHEMATICA
sng[n_]:=Module[{k=1}, While[PrimeOmega[PolygonalNumber[n, k]]!=n, k++]; PolygonalNumber[ n, k]]; Array[sng, 21, 2] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Feb 19 2023 *)
a[n_] := Block[{b = 4 -n, c = n -2, d = n +1, k = 2}, While[ PrimeOmega[k] + PrimeOmega[b + k*c] != d, k++]; PolygonalNumber[n, k]]; Array[a, 19, 2] (* Robert G. Wilson v, Jan 22 2026 *)
PROG
(PARI) a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(n*k - n - 2*k + 4))\2); if(bigomega(t) == n, return(t))); \\ Daniel Suteu, Dec 04 2022
(PARI)
bigomega_polygonals(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, my(t=m*q); if(ispolygonal(t, k), listput(list, t))), forprime(q = p, sqrtnint(B\m, n), my(t=m*q); if(ceil(A/t) <= B\t, list=concat(list, f(t, q, n-1))))); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(k < 3, return()); my(x=2^n, y=2*x); while(1, my(v=bigomega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 04 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 03 2022
EXTENSIONS
a(23)-a(28) from Daniel Suteu, Dec 04 2022
a(2)=4 prepended by Robert Israel, Jan 15 2023
STATUS
approved