A034884 -id:A034884 - cp's OEIS frontend

A056761 Odd numbers less than the cube of their number of divisors.

3, 5, 7, 9, 15, 21, 25, 27, 33, 35, 39, 45, 51, 55, 57, 63, 75, 81, 99, 105, 117, 135, 147, 153, 165, 171, 175, 189, 195, 207, 225, 231, 255, 273, 285, 297, 315, 345, 351, 357, 375, 385, 399, 405, 429, 435, 441, 455, 459, 465, 483, 495, 525, 567, 585, 675, 693

Offset: 1

Labos Elemer, Aug 16 2000

Last term is a(267) = 883575, confirming the author's conjecture. - Charles R Greathouse IV, Apr 27 2011

			14175 = 81*25*7 has 30 divisors, and 30^3 = 27000 > 14175.

Zak Seidov, Table of n, a(n) for n = 1..267 (complete sequence)

Cf. A035033-A035035, A034884, A000005, A000265, A056757-A056767, A056781.

Select[Range[1, 10^6 + 1, 2], DivisorSigma[0, #]^3 > # &] (* Michael De Vlieger, Oct 26 2017 *)

isok(n) = (n % 2) && (numdiv(n)^3 > n); \\ Michel Marcus, Dec 19 2013

A067023 Sigma-crowded numbers: n such that d(n)/sigma(n) is larger than d(m)/sigma(m) for all m > n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 42, 48, 60, 72, 80, 84, 90, 96, 120, 126, 144, 168, 180, 210, 240, 252, 280, 288, 300, 360, 420, 432, 480, 504, 540, 560, 600, 630, 720, 840, 900, 1008, 1080, 1260

Offset: 1

Labos Elemer, Jan 09 2002

nonn

Since d(m) < 2*sqrt(m) < 2*sigma(m), we need only test values of m < (2*sigma(n)/d(n))^2.

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

An analog of A066523. Cf. A002182, A000005, A000203, A004394, A034884, A056758.

crowded[n_] := Module[{}, stop=(2/(dovern=DivisorSigma[0, n]/DivisorSigma[1, n]))^2; For[m=n+1, m=dovern, Return[False]]]; True]; Select[Range[1, 13000], crowded]

A225730 Numbers k such that k < d(k)^(22/10), where d(k) is the number of divisors of k.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 168, 180, 192, 198, 200, 204, 210, 216, 220, 224, 228, 234

Offset: 1

T. D. Noe, May 14 2013

Alternatively, we could write k^5 < d(k)^11. The last odd number is a(23) = 45.

T. D. Noe, Table of n, a(n) for n = 1..155 (complete sequence)

Cf. A000005, A034884 (k < d(k)^2), A175495 (k < 2^d(k)), A056757 (k < d(k)^3).

Cf. A225729-A225738, A353448.

t = {}; Do[If[n < DivisorSigma[0, n]^(22/10), AppendTo[t, n]], {n, 10^5}]; t
Select[Range[250],#Harvey P. Dale, Apr 10 2024 *)

for (k=2, 20000, if (k^5 < numdiv(k)^11, print1(k,", "))) \\ Hugo Pfoertner, Apr 25 2023

A056758 Numbers n for which d(n)^3 < n.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 125, 127, 129, 131, 133, 134, 137, 139, 141, 142, 143, 145, 146

Offset: 1

Labos Elemer, Aug 16 2000

The complementary set (d^3 > n) is finite, albeit with very large terms. See also d(n)^2 < n, A035035.

The last number not in this sequence is 27935107200. - Charles R Greathouse IV, Feb 27 2017

			n=254, d(n)=4, d^3 = 64 < 254 so 254 is in the sequence.

Cf. A035033-A035035, A034884.

is(n)=numdiv(n)^3 < n \\ Charles R Greathouse IV, Feb 25 2017

d(n)^3 = A000005(n)^3 < n.

Erroneous linear recurrence signature link deleted by Harvey P. Dale, Jun 25 2021

A056760 Integers with exactly 2 prime divisors such that the cube of the number of divisors exceeds the number.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172

Offset: 1

Labos Elemer, Aug 16 2000

Numbers with 8 prime divisors also occur among cases satisfying relation d^3>n.

Prime divisors are counted without multiplicity. - Harvey P. Dale, May 14 2012

			The sequence is finite and almost surely complete. Between 270000 and 17000000 no more cases were found. The last 3 entries are: 165888, 186624, 248832. E.g. k = 1024*343 = 248832, with 66 divisors and d^3 = 287496 > 248832.

Donovan Johnson, Table of n, a(n) for n = 1..254 (complete sequence)

Cf. A000005, A033033-A033035, A034884.

Select[Range[180],PrimeNu[#]==2&&DivisorSigma[0,#]^3>#&] (* Harvey P. Dale, May 14 2012 *)

Integers k = (p^w)*(q^u) such that d(k)^3 > k, where d(k) = A000005(k).

A146567 Numbers n such that n*sigma_0(n)/(n+sigma_0(n)) = c, c an integer.

Original entry on oeis.org

2, 12, 24, 56, 60, 132, 1260

Offset: 1

Ctibor O. Zizka, Nov 01 2008

A000027(n)*A000005(n)/(A000027(n)+A000005(n))=c, c an integer.
No other term < 5000000. - Emeric Deutsch, Nov 09 2008
No other term < 10000000. - Michel Marcus, Jun 02 2013
For a given n let x be the minimal natural number such that n*x/(n+x)=c. I conjecture: from a certain n onward, x>sigma_0(n) for all n. Thus, there is no other solution bigger than 1260, and this sequence is finite. - Ctibor O. Zizka, Sep 13 2015
This sequence is complete. The finiteness proof is analogous to that of A152491, after observing that sigma_0(n)^2 < n for n > 1260 (see A034884). - Giovanni Resta, Sep 13 2015

Cf. A000027, A000005, A034884.

with(numtheory): a:=proc (n) if type(n*tau(n)/(n+tau(n)), integer) = true then n else end if end proc: seq(a(n),n=1..200000); # Emeric Deutsch, Nov 09 2008

Corrected and extended (one term) by Emeric Deutsch, Nov 09 2008

A175494 a(n) = floor(n^(1/d(n))), where d(n) = number of divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 4, 1, 2, 2, 4, 1, 2, 2, 2, 1, 5, 1, 5, 1, 2, 2, 2, 1, 6, 2, 2, 1, 6, 1, 6, 1, 1, 2, 6, 1, 3, 1, 2, 1, 7, 1, 2, 1, 2, 2, 7, 1, 7, 2, 1, 1, 2, 1, 8, 2, 2, 1, 8, 1, 8, 2, 2, 2, 2, 1, 8, 1, 2, 3, 9, 1, 3, 3, 3

Offset: 1

Leroy Quet, May 30 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A034884, A175495.

Table[Floor[n^(1/DivisorSigma[0, n])], {n, 100}] (* T. D. Noe, May 14 2013 *)

More terms from R. J. Mathar, May 31 2010

A225737 Numbers n such that n < d(n)^(29/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110

Offset: 1

T. D. Noe, May 14 2013

Alternatively, we could write n^10 < d(n)^29. The last odd term is a(6362) = 225225.

T. D. Noe, Table of n, a(n) for n = 1..21466 (complete sequence)

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).

Cf. A225729-A225738.

t = {}; Do[If[n < DivisorSigma[0, n]^(29/10), AppendTo[t, n]], {n, 10^7}]; t

A225731 Numbers n such that n < d(n)^(23/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192

Offset: 1

T. D. Noe, May 14 2013

Alternatively, we could write n^10 < d(n)^23. The last odd number is a(44) = 105.

T. D. Noe, Table of n, a(n) for n = 1..284 (complete sequence)

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).

Cf. A225729-A225738.

t = {}; Do[If[n < DivisorSigma[0, n]^(23/10), AppendTo[t, n]], {n, 10^5}]; t

A225732 Numbers n such that n < d(n)^(24/10), where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140

Offset: 1

T. D. Noe, May 14 2013

Alternatively, we could write n^5 < d(n)^12. The last odd number is a(95) = 315.

T. D. Noe, Table of n, a(n) for n = 1..528 (complete sequence)

Cf. A034884 (n < d(n)^2), A175495 (n < 2^d(n)), A056757 (n < d(n)^3).

Cf. A225729-A225738.

t = {}; Do[If[n < DivisorSigma[0, n]^(24/10), AppendTo[t, n]], {n, 10^6}]; t

cp's OEIS Frontend

A056761 Odd numbers less than the cube of their number of divisors.

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A067023 Sigma-crowded numbers: n such that d(n)/sigma(n) is larger than d(m)/sigma(m) for all m > n.

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A225730 Numbers k such that k < d(k)^(22/10), where d(k) is the number of divisors of k.

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A056758 Numbers n for which d(n)^3 < n.

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A056760 Integers with exactly 2 prime divisors such that the cube of the number of divisors exceeds the number.

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A146567 Numbers n such that n*sigma_0(n)/(n+sigma_0(n)) = c, c an integer.

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A175494 a(n) = floor(n^(1/d(n))), where d(n) = number of divisors of n.

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A225737 Numbers n such that n < d(n)^(29/10), where d(n) is the number of divisors of n.

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