A087134 -id:A087134 - cp's OEIS frontend

A119317 n-th divisor of A087134(n), the smallest number whose n-th divisor is prime (for n > 1).

1, 2, 3, 5, 7, 7, 13, 13, 19, 17, 23, 31, 29, 31, 41, 61, 61, 61, 47, 61, 73, 97, 127, 71, 97, 107, 127, 149, 173, 211, 127, 97, 107, 211, 127, 149, 173, 181, 211, 257, 281, 317, 181, 421, 241, 257, 281, 317, 337, 367, 421, 509, 563, 631, 727, 421, 487, 509

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A119318, A119319.

a[1] = 1; a[n_] := Module[{k = 1, d}, While[DivisorSigma[0, k] < n || ! PrimeQ[d = Divisors[k][[n]]], k++]; d]; Array[a, 30]  (* Amiram Eldar, Feb 06 2019 *)

A119317(n) := block(
   if equal(n,1) then
      return(1),
   for i : 1 do (
      idv : listify(divisors(i)),
      if length(idv) >= n then (
         if primep(idv[n]) then
            return(idv[n])
        )
     )
)$ /* R. J. Mathar, Mar 13 2012 */

More terms from Amiram Eldar, Feb 06 2019

A119313 Numbers with a prime as third-smallest divisor.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 21, 22, 24, 26, 30, 33, 34, 35, 36, 38, 39, 42, 45, 46, 48, 50, 51, 54, 55, 57, 58, 60, 62, 63, 65, 66, 69, 70, 72, 74, 75, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 98, 102, 105, 106, 108, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

m is a term iff A001221(m) > 1 and (A067029(m) = 1 or A119288(m) < A020639(m)^2).

			a(1) = A087134(3) = 6.
From _Gus Wiseman_, Oct 19 2019: (Start)
The sequence of terms together with their divisors begins:
    6: {1,2,3,6}
   10: {1,2,5,10}
   12: {1,2,3,4,6,12}
   14: {1,2,7,14}
   15: {1,3,5,15}
   18: {1,2,3,6,9,18}
   21: {1,3,7,21}
   22: {1,2,11,22}
   24: {1,2,3,4,6,8,12,24}
   26: {1,2,13,26}
   30: {1,2,3,5,6,10,15,30}
   33: {1,3,11,33}
   34: {1,2,17,34}
   35: {1,5,7,35}
   36: {1,2,3,4,6,9,12,18,36}
   38: {1,2,19,38}
   39: {1,3,13,39}
   42: {1,2,3,6,7,14,21,42}
   45: {1,3,5,9,15,45}
   46: {1,2,23,46}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A119314.
Subsequences: A006881, A000469, A008588.
A subset of A002808 and A080257.
Numbers whose third-largest divisor is prime are A328338.
Second-smallest divisor is A020639.
Third-smallest divisor is A292269.
Cf. A000005, A000040, A001221, A020639, A027750, A033676, A060775, A067029, A088725, A119288, A328189.

q:= n-> (l-> nops(l)>2 and isprime(l[3]))(
         sort([numtheory[divisors](n)[]])):
select(q, [$1..200])[];  # Alois P. Heinz, Oct 19 2019

Select[Range[100],Length[Divisors[#]]>2&&PrimeQ[Divisors[#][[3]]]&] (* Gus Wiseman, Oct 15 2019 *)
Select[Range[130], Length[f = FactorInteger[#]] > 1 && (f[[1, 2]] == 1 || f[[1, 1]]^2 > f[[2, 1]]) &] (* Amiram Eldar, Jul 02 2022 *)

Name edited by Gus Wiseman, Oct 19 2019

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.

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.

A384232 Smallest number whose largest odd noncomposite divisor is its n-th divisor.

Original entry on oeis.org

1, 3, 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, 708120

Offset: 1

Omar E. Pol, May 23 2025

nonn

This coincide with A087134 except for the second term because here a(2) = 3 and there A087134(2) = 2.

			The divisors of 42 are [1, 2, 3, 6, 7, 14, 21, 42] and the largest odd noncomposite divisor is 7 and 7 is its 5th divisor, so a(5) = 42 because 42 the smallest number having that property.

Row 1 of A384233.
Companion of A383402.
Cf. A006005, A027750, A087134.

With[{t = Table[FirstPosition[Divisors[n], FactorInteger[n/2^IntegerExponent[n, 2]][[-1, 1]]][[1]], {n, 1, 10^6}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, May 23 2025 *)

A384233 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd noncomposite divisor is its k-th divisor, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 10, 20, 16, 9, 12, 28, 42, 32, 11, 14, 30, 60, 84, 64, 13, 15, 40, 66, 132, 156, 128, 17, 18, 44, 78, 168, 204, 312, 256, 19, 21, 52, 88, 198, 228, 408, 684, 512, 23, 22, 56, 102, 210, 264, 456, 696, 1020, 1024, 25, 24, 68, 104, 220, 276, 468, 744, 1140, 1380

Offset: 1

Omar E. Pol, May 22 2025

This is a permutation of the positive integers.

			The corner 15 X 15 of the square array is as follows:
      1,  3,  6,  20,  42,  84, 156, 312,  684, 1020, 1380, 1860, 3480, 3720,  4920, ...
      2,  5, 10,  28,  60, 132, 204, 408,  696, 1140, 1740, 2220, 3660, 4440,  5160, ...
      4,  7, 12,  30,  66, 168, 228, 456,  744, 1332, 2040, 2460, 4020, 5580,  5640, ...
      8,  9, 14,  40,  78, 198, 264, 468,  780, 1368, 2088, 2580, 4140, 6960,  6360, ...
     16, 11, 15,  44,  88, 210, 276, 510,  816, 1392, 2232, 2664, 4260, 7224,  6660, ...
     32, 13, 18,  52, 102, 220, 330, 552,  828, 1476, 2280, 2760, 4380, 7632,  7080, ...
     64, 17, 21,  56, 104, 234, 342, 570,  888, 1488, 2436, 2820, 4740, 7896,  7380, ...
    128, 19, 22,  68, 110, 252, 348, 612,  912, 1548, 2544, 2952, 4872, 8280,  7440, ...
    256, 23, 24,  70, 114, 260, 372, 624,  930, 1560, 2604, 3096, 4980, 8496,  7740, ...
    512, 25, 26,  76, 120, 272, 390, 660,  936, 1656, 2736, 3180, 5208, 8784,  8880, ...
   1024, 27, 33,  80, 126, 304, 396, 690,  984, 1692, 2790, 3384, 5220, 8904,  9912, ...
   2048, 29, 34,  90, 130, 306, 414, 792, 1032, 1710, 2832, 3420, 5256, 9030, 10248, ...
   4096, 31, 35,  92, 136, 336, 420, 870, 1044, 1776, 2928, 3540, 5328, 9324, 10440, ...
   8192, 37, 36,  99, 138, 340, 440, 920, 1104, 1908, 3060, 3612, 5340, 9648, 10512, ...
  16384, 41, 38, 100, 140, 368, 444, 966, 1110, 1932, 3108, 3816, 5520, 9660, 10836, ...
  ...
The divisors of 42 are [1, 2, 3, 6, 7, 14, 21, 42] and the largest odd noncomposite divisor is 7 and 7 is its 5th divisor, so T(1,5) = 42 because 42 the smallest number having that property.

Index entries for sequences that are permutations of the natural numbers.

Companion of A383961.
Row 1 gives A384232.
Column 1 gives A000079.
Cf. A006005, A027750, A061345, A065091, A087134.

f[n_] := FirstPosition[Divisors[n], FactorInteger[n/2^IntegerExponent[n, 2]][[-1, 1]]][[1]]; seq[m_] := Module[{t = Table[0, {m}, {m}], v = Table[0, {m}], c = 0, k = 1, i, j}, While[c < m*(m + 1)/2, i = f[k]; If[i <= m, j = v[[i]] + 1; If[j <= m - i + 1, t[[i]][[j]] = k; v[[i]]++; c++]]; k++]; Table[t[[j]][[i - j + 1]], {i, 1, m}, {j, 1, i}] // Flatten]; seq[11] (* Amiram Eldar, May 23 2025 *)

Conjecture: T(n,2) = A061345(n).

A087133 Number of divisors of n that are not greater than the greatest prime-factor of n; a(1)=1.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 4, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 2, 5, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 3, 2, 5, 2, 3, 3, 2, 3, 5, 2, 4, 3, 4, 2, 3, 2, 3, 3, 4, 3, 5, 2, 4, 2, 3, 2, 6, 3, 3, 3, 5, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 4, 4

Offset: 1

Reinhard Zumkeller, Aug 17 2003

nonn

For n > 1, a(n) is the index of the greatest prime factor of n among the divisors of n (see A027750). - Michel Marcus, Jan 21 2019

			n=28: gpf(28)=7 and divisors = {1,2,4,7,14,28}: 1<=7, 2<=7, 4<=7 and 7<=7, therefore a(28)=4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A000005, A006530, A027750, A000961, A087134.

Table[Count[Divisors[n],?(#<=FactorInteger[n][[-1,1]]&)],{n,100}] (* _Harvey P. Dale, May 01 2016 *)

a(n) = if (n==1, 1, my(f = factor(n), gpf = f[#f~,1]); sumdiv(n, d, d <= gpf)); \\ Michel Marcus, Sep 21 2014

a(n) = if (n==1, 1, vecsearch(divisors(n), vecmax(factor(n)[,1]))); \\ Michel Marcus, Jan 21 2019

a(n) <= A000005(n), a(n)=A000005(n) iff n is prime or n=1;
a(n)=2 iff n > 1 is a prime power (A000961);
a(A087134(n))=n and a(k) < n for k < A087134(n).

cp's OEIS Frontend

A119317 n-th divisor of A087134(n), the smallest number whose n-th divisor is prime (for n > 1).

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A119313 Numbers with a prime as third-smallest divisor.

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

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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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A384232 Smallest number whose largest odd noncomposite divisor is its n-th divisor.

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A384233 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd noncomposite divisor is its k-th divisor, n >= 1, k >= 1.

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A087133 Number of divisors of n that are not greater than the greatest prime-factor of n; a(1)=1.

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