A119318
n-th divisor of A119311(n), the smallest number such that the n-th divisor is a prime power.
Original entry on oeis.org
1, 2, 4, 8, 16, 8, 27, 16, 27, 32, 27, 64, 27, 128, 32, 256, 81, 128, 64, 1024, 256, 128, 128, 512, 243, 256, 256, 128, 128, 512, 512, 128, 256, 128, 256, 243, 343, 729, 729, 512, 512, 256, 4096, 729, 1024, 729, 1024, 512, 256, 512, 729, 2048, 256, 1024, 1024
Offset: 1
n=37: the set of divisors of A119311(37) = 41160 has
tau(41160) = 64 elements: {1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 49, 56, 60, 70, 84, 98, 105, 120, 140, 147, 168, 196, 210, 245, 280, 294, [[343]], 392, 420, 490, 588, 686, ...}; the 37th divisor of 41160 = a(37) = 343 = 7^3.
A087134
Smallest number having exactly n divisors that are not greater than the number's greatest prime factor.
Original entry on oeis.org
1, 2, 6, 20, 42, 84, 156, 312, 684, 1020, 1380, 1860, 3480, 3720, 4920, 7320, 10980, 14640, 16920, 21960, 26280, 34920, 45720, 59640, 69840, 89880, 106680, 125160, 145320, 177240, 213360, 244440, 269640, 354480, 320040, 375480, 435960, 456120, 531720, 647640
Offset: 1
See
A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.
-
With[{s = Array[Function[{d, p}, LengthWhile[d, # < p &]] @@ {#, SelectFirst[Reverse@ #, PrimeQ]} &@ Divisors@ # &, 10^6]}, Array[FirstPosition[s, #][[1]] &, Max@ s + 1, 0]] (* Michael De Vlieger, Jan 23 2019 *)
-
a087133(n) = if (n==1, 1, my(f = factor(n), gpf = f[#f~,1]); sumdiv(n, d, d <= gpf));
a(n) = my(k = 1); while (a087133(k) != n, k++); k; \\ Michel Marcus, Sep 21 2014
A221647
Smallest number k such that prime(n) is the n-th divisor of k.
Original entry on oeis.org
3, 10, 28, 66, 234, 204, 456, 828, 1392, 2232, 2220, 5904, 7224, 5640, 9540, 14160, 14640, 28140, 25560, 26280, 79632, 89640, 64080, 69840, 181800, 129780, 134820, 183120, 189840, 213360, 495180, 460320, 934080, 1001280, 380520, 1243440, 1779960
Offset: 2
a(6) = 234 because the divisors of 234 are {1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234}, and prime(6) = 13 is the 6th divisor of 234.
Sequences giving smallest number whose n-th divisor satisfies other conditions:
A087134 (prime),
A119311 (prime power),
A119312 (squarefree),
A226101 (multiple of n-th prime),
A256605 (is n+1),
A383402 (largest odd divisor).
-
A221647 := proc(n)
local p,k,j ;
p := ithprime(n) ;
for j from 1 do
k := j*p ;
dvs := sort(convert(numtheory[divisors](k),list)) ;
if nops(dvs) >= n then
if op(n,dvs) = p then
return k ;
end if;
end if;
end do:
end proc:
seq(A221647(n),n=2..30) ; # R. J. Mathar, May 05 2013
-
nn = 20; t = Table[0, {nn}]; found = 1; n = 2; While[found < nn, n++; d = Divisors[n]; Do[If[i <= nn && d[[i]] == Prime[i] && t[[i]] == 0, t[[i]] = n; found++], {i, Length[d]}]]; Rest[t] (* T. D. Noe, May 07 2013 *)
-
a(n) = my(k=2, f=factor(k), p=prime(n)); while ((numdiv(f)Michel Marcus, May 28 2025
A119312
Smallest number having at least n divisors and a squarefree n-th divisor.
Original entry on oeis.org
1, 2, 6, 6, 12, 30, 30, 30, 60, 90, 60, 210, 120, 210, 210, 210, 420, 630, 420, 630, 420, 630, 420, 1260, 840, 1320, 1260, 1680, 840, 2310, 1260, 2310, 1680, 2640, 5040, 3360, 5280, 2520, 4620, 5460, 4620, 6930, 4620, 6930, 4620, 6930, 4620, 9240, 10920
Offset: 1
See
A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.
Showing 1-4 of 4 results.
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