A383402 -id:A383402 - cp's OEIS frontend

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

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

A383961 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd divisor is its k-th divisor, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 15, 16, 11, 10, 20, 18, 32, 13, 12, 21, 50, 36, 64, 17, 14, 27, 81, 45, 30, 128, 19, 22, 28, 88, 63, 42, 105, 256, 23, 24, 33, 98, 75, 54, 135, 60, 512, 29, 25, 35, 104, 99, 66, 165, 84, 120, 1024, 31, 26, 39, 136, 117, 70, 189, 108, 140, 90

Offset: 1

Omar E. Pol, May 16 2025

This is a permutation of the positive integers.
From Peter Munn, May 18 2025: (Start)
Numbers with the same factorization pattern of their sequence of divisors (see A290110) and the same parity appear here in the same column.
For example, each column k > 2 includes the subsequence 2^(k-2) * p for all prime p > 2^(k-2).
(End)

			The corner 15 X 15 of the square array is as follows:
      1,  3,  6,  15,  18,  36,  30, 105,  60, 120,  90, 315,  816, 1360, 180, ...
      2,  5,  9,  20,  50,  45,  42, 135,  84, 140, 126, 324,  880, 1520, 210, ...
      4,  7, 10,  21,  81,  63,  54, 165, 108, 168, 150, 432,  912, 1632, 252, ...
      8, 11, 12,  27,  88,  75,  66, 189, 132, 220, 198, 440, 1040, 1760, 270, ...
     16, 13, 14,  28,  98,  99,  70, 195, 156, 240, 216, 495, 1056, 1824, 300, ...
     32, 17, 22,  33, 104, 117,  72, 200, 162, 260, 234, 520, 1104, 1840, 330, ...
     64, 19, 24,  35, 136, 147,  78, 231, 204, 308, 264, 525, 1120, 1904, 378, ...
    128, 23, 25,  39, 152, 153, 100, 255, 225, 340, 280, 528, 1144, 2000, 390, ...
    256, 29, 26,  40, 176, 171, 102, 273, 228, 364, 294, 560, 1232, 2080, 396, ...
    512, 31, 34,  44, 184, 175, 110, 285, 276, 380, 306, 585, 1248, 2128, 462, ...
   1024, 37, 38,  51, 208, 207, 114, 297, 348, 405, 312, 616, 1392, 2208, 468, ...
   2048, 41, 46,  52, 232, 243, 130, 345, 372, 460, 336, 624, 1456, 2288, 510, ...
   4096, 43, 48,  55, 242, 245, 138, 351, 400, 476, 342, 675, 1458, 2320, 546, ...
   8192, 47, 49,  56, 248, 261, 144, 357, 441, 480, 350, 680, 1488, 2464, 570, ...
  16384, 53, 58,  57, 296, 272, 154, 375, 444, 500, 408, 693, 1496, 2480, 588, ...
  ...

Column 1 gives A000079.
Column 2 gives A065091.
Column 3 consists of (A001248 U A091629 U A100484)\{4}.
Column 4 consists of numbers >= 15 in (A001749 U A030078 U A046388 U A070875).
Row 1 gives A383402.
Cf. A000005, A000265, A001227, A027750, A038547, A174090, A182469, A290110, A383401.

f[n_] := If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[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 16 2025 *)

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 *)

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

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A383961 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd divisor is its k-th divisor, n >= 1, k >= 1.

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

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Mathematica