A358863 a(n) is the smallest n-gonal number with exactly n prime factors (counted with multiplicity).
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
Keywords
Examples
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.
Links
- Eric Weisstein's World of Mathematics, Prime Factor
- Eric Weisstein's World of Mathematics, Polygonal Number
Programs
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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 *) -
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
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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
Formula
A001222(a(n)) = n. - Robert Israel, Jan 15 2023
Extensions
a(23)-a(28) from Daniel Suteu, Dec 04 2022
a(2)=4 prepended by Robert Israel, Jan 15 2023
Comments