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A358863
a(n) is the smallest n-gonal number with exactly n prime factors (counted with multiplicity).
7
4, 28, 16, 176, 4950, 8910, 1408, 346500, 277992, 7542080, 326656, 544320, 120400000, 145213440, 48549888, 4733575168, 536813568, 2149576704, 3057500160, 938539560960, 1358951178240, 36324805836800, 99956555776, 49212503949312, 118747221196800, 59461613912064, 13749193801728
OFFSET
2,1
COMMENTS
The corresponding indices of n-gonal numbers are 7, 4, 11, 50, 60, 22, 315, 264, 1295, 256, 315, 4480, 4727, 2634, 25123, 8192, 15903, 18432, 314315, 368640, ...
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
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 *)
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
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