A221647 -id:A221647 - cp's OEIS frontend

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

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

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.

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

Reinhard Zumkeller, May 15 2006

nonn

Cf. A005117, A087134, A119311.

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

A383402 Smallest number whose largest odd divisor is its n-th divisor.

Original entry on oeis.org

1, 3, 6, 15, 18, 36, 30, 105, 60, 120, 90, 315, 816, 1360, 180, 700, 450, 360, 720, 1008, 420, 1540, 630, 900, 840, 1080, 1620, 1680, 2160, 1800, 1890, 5280, 1260, 3240, 3150, 17325, 7200, 29120, 5670, 9072, 2520, 3960, 10296, 18144, 3780, 20020, 5040, 7920, 10800

Offset: 1

Omar E. Pol, May 14 2025

nonn

From Peter Munn, May 15 2025 and May 20 2025: (Start)
A038547 is easily seen to be an upper bound for the sequence and a term equals this upper bound if and only if it is odd. Moreover, if a(n) = 2m with m odd, then the largest odd divisor of 2m is m, its second largest divisor, and a(n) = 2 * A038547((n+1)/2). It follows that 1 is the only term not divisible by 4 or by a nonunit term of A038547.
a(8) = 105 is the last squarefree term. (This is a corollary to lemma: prime p > 9 cannot be a divisor of a squarefree term. Proof of lemma: Let p divide squarefree k. If 3p is also divisor, set m = 9k/p, otherwise set m = 3k/p. Then k is not a term as m is a smaller number whose largest odd divisor is in the same position in the divisor list.)
(End)
If a(n) = m then m has at least n divisors. - David A. Corneth, May 16 2025
Every term a(n) = t > 1 is divisible by 2 or 3. Proof: Suppose it is not. Then it is odd and n is the number of divisors of t (cf. A000005). But t is not the smallest number that has n odd divisors that is odd. Setting every prime factor p to the largest prime < p and then multiplying gives a smaller odd number that has n divisors (cf. A064989). - David A. Corneth, May 17 2025

			The divisors of 18 are [1, 2, 3, 6, 9, 18] and the largest odd divisor is 9 and 9 is its 5th divisor, so a(5) = 18 because 18 the smallest number having that property.

David A. Corneth, Table of n, a(n) for n = 1..872
David A. Corneth, PARI program
David A. Corneth, Upper bounds on a(n) for n = 1..10000
Michael De Vlieger, Prime Power Decomposition of A383402(n), n = 1..261.

Row 1 of A383961.
The range of terms is a subset of {1} U A355200.
Cf. A000005, A000265, A001227, A005117, A027750, A038547, A064989, A182469, A383401.
See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.

With[{t = Table[If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[1]]], {n, 1, 30000}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, May 14 2025 *)

a(n) = my(k=1); while (select(x->(x==k/2^valuation(k,2)), divisors(k), 1)[1] != n, k++); k; \\ Michel Marcus, May 14 2025

PARI
```
\\ See Corneth link
```

a(n) = min({k : A000005(k) >= n & A027750(k,n) = A000265(k)}). - Peter Munn, May 14 2025

More terms from Amiram Eldar, May 14 2025

A226101 Smallest number k such that prime(n) divides the n-th divisor of k.

Original entry on oeis.org

3, 10, 14, 44, 52, 102, 114, 276, 348, 372, 444, 984, 1032, 1128, 1272, 2124, 2196, 3216, 3408, 4380, 4740, 4980, 5340, 11640, 12120, 12360, 12840, 13080, 13560, 15240, 15720, 24660, 25020, 26820, 27180, 37680, 39120, 40080, 41520, 64440, 65160, 68760, 69480

Offset: 2

Michel Lagneau, May 26 2013

nonn

			a(9) = 276 because the divisors of 276 are {1, 2, 3, 4, 6, 12, 23, 46, 69, 92, 138, 276}, and prime(9) = 23, the 9th divisor of 276 is 69 and 23 | 69.

Cf. A000040, A027750, A221647.

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

nn = 45; t = Table[0, {nn}]; found = 1; n = 2; While[found < nn, n++; d = Divisors[n]; Do[If[i <= nn && Mod[d[[i]],Prime[i]]==0 && t[[i]] == 0, t[[i]] = n; found++], {i, Length[d]}]]; Rest[t] (* Program from T. D. Noe, adapted for this sequence (see A221647) *)

A256605 Least k such that n+1 is the n-th divisor of k.

Original entry on oeis.org

3, 4, 20, 12, 84, 120, 360, 360, 3960, 2520, 32760, 27720, 27720, 55440, 942480, 720720, 13693680, 12252240, 12252240, 12252240, 281801520, 232792560, 1163962800, 1163962800, 3491888400, 3491888400, 101264763600, 80313433200, 2489716429200, 4658179125600

Offset: 2

Michel Lagneau, Apr 04 2015

The case n = 1 is not possible because the number 2 is never the first divisor of k (1 is the first divisor).

			a(6) = 84 because the divisors of 84 are {1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84} and 7 is the 6th divisor of 84.

Cf. A003418, A007917, A225562.

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

with(numtheory):for n from 2 to 31 do:ii:=0:for  k from 1 to 10^9 while(ii=0) do:x:=divisors(k):n1:=nops(x):if n<=n1 and x[n]=n+1 then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:

nn=20;t=Table[0,{nn}];found=1;n=2;While[found

a(n) = {k = 1; ok = 0; while (!ok, d = divisors(k); if ((#d >= n) && (d[n] == n+1), ok = 1, k++);); k;} \\ Michel Marcus, Apr 04 2015

a(n) = A003418(n+1)/A007917(n). - Peter Munn, May 14 2025

A233933 Smallest number k such that R(n) is the n-th divisor of k, where R(n) is the n-th Ramanujan prime (A104272).

Original entry on oeis.org

11, 34, 116, 246, 752, 708, 4288, 1704, 3492, 4848, 11556, 7620, 28608, 47112, 24048, 21480, 45612, 40860, 54960, 218088, 180684, 121464, 94680, 242100, 269760, 486288, 313488, 249840, 376920, 308280, 738540, 721800, 515340, 1106160, 930960, 935280, 737520

Offset: 2

Michel Lagneau, Dec 18 2013

nonn

			a(2) = 11 because the divisors of 11 are {1, 11}, and the 2nd divisor of 11 is 11 = A104272(2);
a(3) = 34 because the divisors of 34 are {1, 2, 17, 34}, and the 3rd divisor of 34 is 17 = A104272(3).

Cf. A104272, A221647.

nn=20; R=Table[0,{nn}]; s=0; Do[If[PrimeQ[k],s++]; If[PrimeQ[k/2],s--]; If[sT. D. Noe adapted for this sequence - see A104272 and A221647 *)

cp's OEIS Frontend

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

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

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A119312 Smallest number having at least n divisors and a squarefree n-th divisor.

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A383402 Smallest number whose largest odd divisor is its n-th divisor.

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A226101 Smallest number k such that prime(n) divides the n-th divisor of k.

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A256605 Least k such that n+1 is the n-th divisor of k.

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Maple

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A233933 Smallest number k such that R(n) is the n-th divisor of k, where R(n) is the n-th Ramanujan prime (A104272).

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