A119312 -id:A119312 - cp's OEIS frontend

A119319 n-th divisor of A119312(n), the smallest number such that the n-th divisor is squarefree.

1, 2, 3, 6, 6, 10, 15, 30, 15, 30, 30, 35, 30, 70, 105, 210, 42, 70, 70, 105, 105, 210, 210, 70, 105, 165, 105, 105, 210, 770, 210, 2310, 210, 330, 105, 210, 330, 210, 385, 455, 462, 770, 770, 1155, 1155, 2310, 2310, 385, 455, 330, 462, 455, 330, 770, 910, 455

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

Cf. A005117, A119317, A119318.

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

Reinhard Zumkeller, Aug 17 2003

nonn

A087133(a(n))=n.
Also smallest number such that the n-th divisor is prime. - Reinhard Zumkeller, May 15 2006
From David A. Corneth, Jan 22 2019: (Start)
For the first 10000 terms except 1, a(n) is of the form A025487(k) * p where p is the smallest prime larger than the n-th divisor and, if the (n+1)-th divisor exists, less than that divisor.
This sequence isn't a sequence of indices of records to A087133 as it's not monotonically increasing; 354480 = a(34) > a(35) = 320040. (End)

			a(3) = A119313(1) = 6.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A006530, A025487, A087133, A119311, A119312.

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

More terms from Reinhard Zumkeller, May 15 2006

More terms from 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

Michel Lagneau, May 04 2013

nonn

The similar problem "smallest number k such that prime(n) is the n-th prime divisor of k" is given by the sequence A002110: primorial numbers product of first n primes.

			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.

Donovan Johnson, Table of n, a(n) for n = 2..300

Cf. A000040, A002110, A027750.

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

A119311 Smallest number having at least n divisors and a prime power as n-th divisor.

Original entry on oeis.org

1, 2, 4, 8, 16, 24, 54, 48, 108, 96, 216, 192, 540, 384, 480, 768, 1296, 1152, 960, 3072, 2304, 2688, 1920, 4608, 4860, 5376, 3840, 8064, 5760, 10752, 7680, 17280, 19200, 13440, 11520, 19440, 41160, 40824, 29160, 32256, 23040, 26880, 61440, 81648

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

Cf. A000961, A087134, A119312.

See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.

cp's OEIS Frontend

A119319 n-th divisor of A119312(n), the smallest number such that the n-th divisor is squarefree.

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A087134 Smallest number having exactly n divisors that are not greater than the number's greatest prime factor.

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A221647 Smallest number k such that prime(n) is the n-th divisor of k.

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Maple

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A119311 Smallest number having at least n divisors and a prime power as n-th divisor.

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