A038547 -id:A038547 - cp's OEIS frontend

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.

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 *)

A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

Original entry on oeis.org

1, 2, 12, 120, 1680, 36960, 960960, 32672640, 1241560320, 57111774720, 3312482933760, 205373941893120, 15197671700090880, 1246209079407452160, 107173980829040885760, 10074354197929843261440, 1067881544980563385712640, 126010022307706479514091520, 15373222721540190500719165440

Offset: 0

Ilya Gutkovskiy, Apr 26 2018

nonn

For n > 0, a(n) is the smallest number m having exactly n distinct prime divisors and exactly 2*n - 1 prime divisors counted with multiplicity.

			a(1) = 2^1;
a(2) = 2^2*3;
a(3) = 2^3*3*5;
a(4) = 2^4*3*5*7;
a(5) = 2^5*3*5*7*11, etc.

Central terms of triangle A303555 (for n > 0).

Cf. A000079, A002110, A003680, A005179, A011782, A038547, A061283, A070175, A088860, A102476.

Join[{1}, Table[2^(n - 1) Product[Prime[j], {j, n}], {n, 18}]]

a(n) = A011782(n)*A002110(n).

A335029 Numbers that are not practical (A237287) and have more divisors than any smaller number that is not practical.

Original entry on oeis.org

3, 9, 10, 44, 70, 225, 315, 770, 1575, 2835, 3465, 10010, 17325, 31185, 45045, 121275, 135135, 225225, 405405, 675675, 1576575, 2027025, 2297295, 3828825, 6891885, 11486475, 26801775, 34459425, 43648605, 72747675, 130945815, 218243025, 509233725, 654729075, 1003917915

Offset: 1

Amiram Eldar, May 20 2020

nonn

The corresponding numbers of divisors are 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 96, 108, 120, 128, 144, 160, 192, 216, 240, 256, 288, 320, 384, 432, 480, 512, ...
Of the first 39 terms, 34 terms are also in A038547.
None of the terms are highly composite (A002182) since all the highly composite numbers are practical numbers (A005153).

			The first 5 numbers that are not practical are 3, 5, 7, 9, 10. Their numbers of divisors are 2, 2, 2, 3, 4. The record numbers of divisors are 2, 3 and 4 which occur at 3, 9 and 10.

Cf. A002182, A005153, A237287, A275239, A335008, A335030.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; seq = {}; dm = 1; Do[fct = FactorInteger[n]; d = Times @@ (1 + Last/@ fct); If[d > dm && !pracQ[fct], dm = d; AppendTo[seq, n]], {n, 3, 10^5}]; seq

A356666 Smallest m such that the m-th Lucas number has exactly n divisors that are also Lucas numbers.

Original entry on oeis.org

1, 0, 3, 6, 15, 30, 45, 90, 105, 210, 405, 810, 315, 630, 3645, 2025, 945, 1890, 1575, 3150, 2835, 5670, 36450, 25025, 3465, 6930, 101250, 11025, 22050, 51030, 14175, 28350, 10395, 20790, 2952450, 175175, 17325, 34650, 1937102445, 625625, 31185, 62370, 127575, 255150

Offset: 1

Michel Marcus, Aug 22 2022

nonn

Further terms <= 51030: a(28) = 11025, a(29) = 22050, a(30) = 51030, a(31) = 14175, a(32) = 28350, a(33) = 10395, a(34) = 20790, a(37) = 17325, a(38) = 34650, a(41) = 31185, a(49) = 45045. - Daniel Suteu, Aug 24 2022

David A. Corneth, Table of n, a(n) for n = 1..382

Cf. A000032, A038547, A102460, A304092, A356123.

Cf. A105802 (similar for Fibonacci).

L(n)=fibonacci(n+1)+fibonacci(n-1); \\ A000032
isld(n) = { my(u1=1, u2=3, old_u1); if(n<=2, sign(n), while(n>u2, old_u1=u1; u1=u2; u2=old_u1+u2); (u2==n)); }; \\ A102460
nbld(n) = sumdiv(n, d, isld(d)); \\ A304092
a(n) = my(k=0); while(nbld(L(k)) != n, k++); k;

countLd(n) = my(c=0,x=2,y=1); while(x<=n, if(n%x==0, c++); [x,y]=[y,x+y]); c;
a(n) = if(n==1, return(1)); my(k=0,x=2,y=1); while(1, if(countLd(x) == n, return(k)); [x,y,k]=[y,x+y,k+1]); \\ Daniel Suteu, Aug 24 2022

A000032(a(n)) = A356123(n).

a(12)-a(26) from Daniel Suteu, Aug 24 2022

More terms from Daniel Suteu and David A. Corneth, Sep 04 2022

A360438 Smallest number with 2^n odd divisors.

Original entry on oeis.org

1, 3, 15, 105, 945, 10395, 135135, 2297295, 43648605, 1003917915, 25097947875, 727840488375, 22563055139625, 834833040166125, 34228154646811125, 1471810649812878375, 69175100541205283625, 3389579926519058897625, 179647736105510121574125, 10599216430225097172873375, 646552202243730927545275875

Offset: 0

Hartmut F. W. Hoft, Feb 07 2023

nonn

			a(4) = A038547(2^4) = 945 = 3^(2^2-1) * 5^(2^1-1) * 7^(2^1-1)  = 3^3 * 5 * 7,
a(5) = A038547(2^5) = 10395 = 3^(2^2-1) * 5^(2^1-1) * 7^(2^1-1) * 11^(2^1-1) = 3^3 * 5 * 7 * 11,
a(24) = 3^3 * 5^3 * 7^3 * 11 * ... *79, and
a(25) = 3^7 * 5^3 * 7^3 * 11 * ... *79 since 79 < 3^4 < 83.

Hartmut F. W. Hoft, Computational Characterization of a(n) = A038547(2^n)

Cf. A005179, A038547.

next[{num_, fList_, lastP_, {p_, k_}}] := Module[{nP, f1List, p1, k1}, nP=NextPrime[First[Last[fList]]]; If[nP{p, 2k-1}, {1}]; {{p1, k1}}=FactorInteger[Min[Map[#[[1]]^(#[[2]]+1)&, f1List]]]; {num p^k, f1List, lastP, {p1, k1}}]]
a360438[n_] := Join[{1}, Map[First, NestList[next, {3, {{3, 1}}, 3, {3, 2}}, n-1]]]/;n>=1
Join[{1}, a360438[20]]

a(n) = A038547(2^n).

A364582 a(n) is the least number with exactly n divisors of the form 3*k+1.

Original entry on oeis.org

1, 4, 16, 28, 80, 112, 320, 280, 784, 560, 1600, 1120, 10000, 2240, 3920, 2800, 25600, 5600, 1310720, 6160, 15680, 11200, 48400, 12320, 110000, 70000, 39200, 24640, 6553600, 30800, 5368709120, 36400, 78400, 179200, 440000, 61600, 343597383680, 1210000, 490000, 80080

Offset: 1

Ilya Gutkovskiy, Jul 28 2023

nonn

Bert Dobbelaere, Table of n, a(n) for n = 1..200

Cf. A001817, A005179, A038547, A364583.

a(n) = my(k=1); while (sumdiv(k, d, (d%3)==1) != n, k++); k; \\ Michel Marcus, Jul 29 2023

More terms from Bert Dobbelaere, Jul 31 2023

A364584 a(n) is the least number with exactly n divisors of the form 4*k+1.

Original entry on oeis.org

1, 5, 25, 45, 441, 225, 5103, 585, 1575, 2205, 35721, 2925, 194481, 25515, 11025, 9945, 2893401, 17325, 2711943423, 28665, 127575, 178605, 480249, 45045, 275625, 972405, 121275, 266805, 18983603961, 143325, 1441237924662543, 135135, 893025, 14467005, 3189375, 225225

Offset: 1

Ilya Gutkovskiy, Jul 28 2023

nonn

Bert Dobbelaere, Table of n, a(n) for n = 1..200

Cf. A001826, A005179, A038547, A364585.

More terms from Bert Dobbelaere, Jul 31 2023

A364585 a(n) is the least number with exactly n divisors of the form 4*k+3.

Original entry on oeis.org

1, 3, 15, 63, 105, 567, 315, 3969, 945, 1575, 2835, 413343, 3465, 53361, 19845, 14175, 10395, 1750329, 17325, 26040609, 31185, 99225, 480249, 219667417263, 45045, 354375, 266805, 121275, 257985, 160137547184727, 155925, 4322241, 135135, 10333575, 8751645, 2480625

Offset: 0

Ilya Gutkovskiy, Jul 28 2023

nonn

Bert Dobbelaere, Table of n, a(n) for n = 0..200

Cf. A001842, A005179, A038547, A364584.

More terms from Bert Dobbelaere, Aug 01 2023

A121858 Smallest odd number having prime(n) divisors, where prime(n) is the n-th prime=A000040(n).

Original entry on oeis.org

3, 9, 81, 729, 59049, 531441, 43046721, 387420489, 31381059609, 22876792454961, 205891132094649, 150094635296999121, 12157665459056928801, 109418989131512359209, 8862938119652501095929, 6461081889226673298932241

Offset: 1

Lekraj Beedassy, Aug 30 2006

nonn

a(n) is also the smallest number k with the property that the symmetric representation of sigma(k) has prime(n) subparts. - Omar E. Pol, Oct 08 2022

Cf. A006093, A061286.

Cf. A000040, A038546, A237593, A279387.

3^(Prime[Range[20]]-1) (* Harvey P. Dale, Mar 19 2013 *)

a(n) = 3^(prime(n)-1) = 3^A006093(n).

a(n) = A038547(A000040(n)). - Omar E. Pol, Oct 08 2022

A287661 Smallest odd number with exactly n nonprime divisors.

Original entry on oeis.org

1, 9, 27, 45, 105, 135, 225, 405, 315, 675, 177147, 1155, 945, 3375, 1575, 6075, 2835, 10125, 18225, 3465, 4725, 30375, 50625, 11025, 25515, 91125, 14175, 10395, 23625, 273375, 1476225, 17325, 33075, 759375, 50031545098999707, 31185, 70875, 1366875, 127575

Offset: 1

Ilya Gutkovskiy, May 29 2017

nonn

			a(5) = 105 because 105 has 8 divisors {1, 3, 5, 7, 15, 21, 35, 105} among which 5 are nonprime {1, 15, 21, 35, 105} and 105 is the smallest odd with exactly 5 nonprime divisors.

Giovanni Resta, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Divisor

Cf. A001221, A002110, A003680, A005179, A033273, A038547, A055079, A070826.

A033273(a(n)) = n.

a(35)-a(39) from Giovanni Resta, May 31 2017

cp's OEIS Frontend

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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A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

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A335029 Numbers that are not practical (A237287) and have more divisors than any smaller number that is not practical.

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A356666 Smallest m such that the m-th Lucas number has exactly n divisors that are also Lucas numbers.

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A360438 Smallest number with 2^n odd divisors.

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A364582 a(n) is the least number with exactly n divisors of the form 3*k+1.

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A364584 a(n) is the least number with exactly n divisors of the form 4*k+1.

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A364585 a(n) is the least number with exactly n divisors of the form 4*k+3.

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A121858 Smallest odd number having prime(n) divisors, where prime(n) is the n-th prime=A000040(n).

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