A051346 -id:A051346 - cp's OEIS frontend

A051520 Duplicate of A051346.

11264, 14175, 28160, 44100, 46464, 51200, 64000, 82944, 95744, 96000, 107008, 109375

Offset: 1

dead

A051278 Numbers n such that n = k/d(k) has a unique solution, where d(k) = number of divisors of k.

4, 6, 9, 10, 12, 14, 15, 20, 21, 22, 26, 32, 33, 34, 35, 36, 38, 39, 42, 46, 50, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 75, 77, 78, 82, 85, 86, 87, 90, 91, 93, 94, 95, 96, 98, 100, 102, 106, 108, 110, 111, 114, 115, 118, 119, 122, 123, 126, 128, 129, 130

Offset: 1

N. J. A. Sloane, R. K. Guy

Because d(k) <= 2*sqrt(k), it suffices to check k from 1 to 4*n^2. - Nathaniel Johnston, May 04 2011

A051521(a(n)) = 1. - Reinhard Zumkeller, Dec 28 2011

			36 is the unique number k with k/d(k)=4.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A033950, A036763, A051279, A051280, A051346.

a051278 n = a051278_list !! (n-1)
a051278_list = filter ((== 1) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011

with(numtheory): A051278 := proc(n) local ct,k: ct:=0: for k from 1 to 4*n^2 do if(n=k/tau(k))then ct:=ct+1: fi: od: if(ct=1)then return n: else return NULL: fi: end: seq(A051278(n),n=1..40);

cnt[n_] := Count[Table[n == k/DivisorSigma[0, k], {k, 1, 4*n^2}], True]; Select[Range[130], cnt[#] == 1 &]  (* Jean-François Alcover, Oct 22 2012 *)

A051279 Numbers n such that n = k/d(k) has exactly 2 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

1, 2, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 28, 29, 31, 37, 41, 43, 44, 47, 48, 52, 53, 56, 59, 61, 67, 68, 71, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 97, 101, 103, 104, 107, 109, 113, 116, 120, 124, 127, 131, 132, 136, 137, 139, 148, 149, 151, 152, 154, 156

Offset: 1

N. J. A. Sloane, R. K. Guy, David W. Wilson

Because d(k) <= 2*sqrt(k), it suffices to check k from 1 to 4*n^2. - Nathaniel Johnston, May 04 2011

A051521(a(n)) = 2. - Reinhard Zumkeller, Dec 28 2011

			There are exactly 2 numbers k, 40 and 60, with k/d(k)=5.

Nathaniel Johnston and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 150 terms from Nathaniel Johnston)

Cf. A033950, A036763, A051278, A051280, A051346.

a051279 n = a051279_list !! (n-1)
a051279_list = filter ((== 2) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011

with(numtheory): A051279 := proc(n) local ct, k: ct:=0: for k from 1 to 4*n^2 do if(n=k/tau(k))then ct:=ct+1: fi: od: if(ct=2)then return n: else return NULL: fi: end: seq(A051279(n), n=1..40); # Nathaniel Johnston, May 04 2011

A051279 = Reap[Do[ct = 0; For[k = 1, k <= 4*n^2, k++, If[n == k/DivisorSigma[0, k], ct++]]; If[ct == 2, Print[n]; Sow[n]], {n, 1, 160}]][[2, 1]](* Jean-François Alcover, Apr 16 2012, after Nathaniel Johnston *)

A051280 Numbers n such that n = k/d(k) has exactly 3 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

3, 25, 40, 49, 54, 121, 125, 135, 140, 169, 189, 216, 220, 250, 260, 289, 297, 340, 351, 361, 375, 380, 400, 459, 460, 500, 513, 529, 580, 620, 621, 675, 729, 740, 770, 783, 820, 837, 841, 860, 875, 882, 910, 940, 961, 999, 1060, 1107, 1152, 1161, 1180, 1188, 1190

Offset: 1

N. J. A. Sloane, R. K. Guy, David W. Wilson

Many terms are of the form a(k) * p^m/(m+1), where p is coprime to the three solutions for k. The sequence of "primitive" terms (i.e. not expressible this way) begins 3, 40, 54, 125, 135, 216, 250.... Are there any such numbers that admit a fourth solution? - Charlie Neder, Feb 13 2019

			There are exactly 3 numbers k, 9, 18 and 24, with k/d(k) = 3.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A033950, A036763, A051278, A051279, A051346.

(Select[Table[k / Length @ Divisors[k], {k, 1, 200000}], IntegerQ] // Sort // Split // Select[#, Length[#] == 3 &] &)[[All, 1]][[1 ;; 53]] (* Jean-François Alcover, Apr 22 2011 *)

A217125 Numbers n such that n = k/d(k) has exactly 4 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

11264, 14175, 28160, 44100, 46464, 51200, 95744, 96000, 107008, 109375, 109760, 116160, 129536, 151263, 162624, 163328, 174592, 192000, 208384, 224000, 230912, 239360, 242176, 242550, 246960, 264704, 267520, 281600, 286650, 298496, 302016, 323840, 332288

Offset: 1

Donovan Johnson, Sep 27 2012

nonn

			k/d(k) = 11264 for exactly 4 k values: 360448, 585728, 630784 and 1115136.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A051278, A051279, A051280, A051346, A217126, A217127.

(* Assuming 3*10^5 <= k <= 3*10^8 *) ClearAll[cnt]; cnt[] = 0; Do[ If[IntegerQ[n = k/DivisorSigma[0, k]], cnt[n]++; If[cnt[n] >= 4, Print[{n, k, cnt[n]}]]], {k, 3*10^5, 3*10^8}]; Select[Range[350000], cnt[#] == 4 &] (* _Jean-François Alcover, Sep 28 2012 *)

A217126 Numbers n such that n = k/d(k) has exactly 5 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

64000, 290304, 352000, 432000, 544000, 608000, 736000, 928000, 992000, 1036800, 1184000, 1312000, 1376000, 1504000, 1512000, 1596672, 1696000, 1888000, 1952000, 2100875, 2144000, 2272000, 2336000, 2467584, 2515968, 2528000, 2592000, 2656000, 2757888, 2848000

Offset: 1

Donovan Johnson, Sep 27 2012

nonn

			k/d(k) = 64000 for exactly 5 k values: 4096000, 6656000, 7040000, 7168000 and 11520000.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A051278, A051279, A051280, A051346, A217125, A217127.

A217127 Numbers n such that n = k/d(k) has exactly 6 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

82944, 456192, 705024, 787968, 953856, 1202688, 1285632, 1534464, 1700352, 1783296, 1949184, 2198016, 2446848, 2529792, 2778624, 2944512, 3027456, 3276288, 3345408, 3442176, 3691008, 3877632, 4022784, 4188672, 4271616, 4333824, 4437504, 4520448, 4686336

Offset: 1

Donovan Johnson, Sep 27 2012

nonn

			k/d(k) = 82944 for exactly 6 k values: 7962624, 12939264, 13271040, 13934592, 21565440 and 23224320.

Donovan Johnson, Table of n, a(n) for n = 1..300

Cf. A051278, A051279, A051280, A051346, A217125, A217126.

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A051520 Duplicate of A051346.

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A051278 Numbers n such that n = k/d(k) has a unique solution, where d(k) = number of divisors of k.

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A051279 Numbers n such that n = k/d(k) has exactly 2 solutions, where d(k) = number of divisors of k.

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A051280 Numbers n such that n = k/d(k) has exactly 3 solutions, where d(k) = number of divisors of k.

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A217125 Numbers n such that n = k/d(k) has exactly 4 solutions, where d(k) = number of divisors of k.

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A217126 Numbers n such that n = k/d(k) has exactly 5 solutions, where d(k) = number of divisors of k.

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A217127 Numbers n such that n = k/d(k) has exactly 6 solutions, where d(k) = number of divisors of k.

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