A034884 -id:A034884 - cp's OEIS frontend

A046525 Numbers common to A033950 and A034884.

2, 8, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 96, 108, 132, 180, 240, 252, 288, 360, 480, 504, 720, 1260

Offset: 1

Select[Range[1265], Mod[#, x = DivisorSigma[0, #]] == 0 && # < x^2 &] (* Jayanta Basu, Jun 27 2013 *)

A056757 Cube of number of divisors is larger than the number.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 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

Labos Elemer, Aug 16 2000

Sequence is finite with 51261 terms. - Charles R Greathouse IV, Apr 27 2011 [Corrected by Amiram Eldar, Jun 02 2024]

The last odd term is a(15199) = 883575. The odd terms are in A056761. - T. D. Noe, May 14 2013

			k = 27935107200 = 128*27*25*7*11*13*17*19 has 3072 divisors, 3072^3/k = 1.03779..., so k is a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..51261 (complete sequence, corrected by Amiram Eldar)

Cf. A000005, A034884, A035033, A035034, A035035, A056761, A066693.

Cf. A175495 (k < 2^d(k)), A225729-A225738.

t = {}; Do[If[n < DivisorSigma[0,n]^3, AppendTo[t, n]], {n, 10^3}]; t (* T. D. Noe, May 14 2013 *)
Select[Range[120],DivisorSigma[0,#]^3>#&] (* Harvey P. Dale, Apr 22 2019 *)

is(n)=numdiv(n)^3>n \\ Charles R Greathouse IV, Sep 14 2015

{ k : A000005(k)^3 > k}.

A035033 Numbers k such that k <= d(k)^2, where d() = number of divisors (A000005).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 108, 120, 126, 132, 140, 144, 168, 180, 192, 210, 216, 240, 252, 288, 300, 336, 360, 420, 480, 504, 540, 720, 840, 1260

Offset: 1

Erich Friedman, Labos Elemer

Cf. A000005, A035034, A035035, A033950, A036763, A034884, A273323.

Select[Range[1300],#<=DivisorSigma[0,#]^2&] (* Harvey P. Dale, Mar 16 2012 *)

is(n)=numdiv(n)^2>=n \\ Charles R Greathouse IV, Aug 03 2012

A035035 Numbers k such that k > d(k)^2, where d(k) is the number of divisors of k.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82

Offset: 1

N. J. A. Sloane

nonn

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

Complement of A035033.

Cf. A000005, A034884, A035034.

Select[Range[100], DivisorSigma[0, #]^2 < # &] (* Amiram Eldar, Aug 29 2020 *)

a(n) = n + 54 for n >= 1207. - Amiram Eldar, Aug 29 2020

A175495 Positive integers k such that k < 2^d(k), where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 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, 144, 150, 152, 154, 156

Offset: 1

Leroy Quet, May 30 2010

nonn

Numbers k for which A175494(k) = 1.
After the initial 1 in this sequence, the first integer in this sequence but not in A034884 is 44.
All 52 terms of A034884 are also in this sequence. - Zak Seidov, May 30 2010
All powers of 2 are terms. - D. S. McNeil, May 30 2010
It follows from the Wiman-Ramanujan theorem that, for every eps > 0 and k > k_0(eps), we have k > tau(k)^(log(log(k))/(log(2)+eps)). Therefore in particular A034884 is finite. On the other hand, for 0 < eps < log(2), it is known that there exist infinitely many numbers for which k < tau(k)^(log(log(k))/(log(2)-eps)), that is, tau(k) > k^((log(2)-eps)/log(log(k))) and 2^tau(k) > 2^(k^((log(2)-eps)/log(log(k)))) >> k. In particular, A175495 is infinite. - Vladimir Shevelev, May 30 2010

K. Prachar, Primzahlverteilung, Springer-Verlag, 1957, Ch. 1, Theorem 5.2.
S. Ramanujan, Highly composite numbers, Collected papers, Cambridge, 1927, 85-86.
A. Wiman, Sur l'ordre de grandeur du nombre de diviseurs d'un entier, Arkiv Mat. Astr. och Fys., 3, no. 18 (1907), 1-9.

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

Cf. A000005, A175494, A034884.

t = {}; n = 0; While[Length[t] < 100, n++; If[n < 2^DivisorSigma[0, n], AppendTo[t, n]]]; t (* T. D. Noe, May 14 2013 *)
Select[Range[200],#<2^DivisorSigma[0,#]&] (* Harvey P. Dale, Apr 24 2015 *)

isok(n) = n < 2^numdiv(n); \\ Michel Marcus, Sep 09 2019

from sympy import divisor_count
def ok(n): return n < 2**divisor_count(n)
print(list(filter(ok, range(1, 157)))) # Michael S. Branicky, Jul 29 2021

More terms from Jon E. Schoenfield, Jun 13 2010

A225729 Numbers n such that n < d(n)^(21/10), 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, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 168, 180, 192, 210, 216, 240, 252, 264, 270, 280, 288, 300, 312, 330, 336, 360, 396

Offset: 1

T. D. Noe, May 14 2013

Alternatively, we could write n^10 < d(n)^21. The last odd number is a(10) = 15.

T. D. Noe, Table of n, a(n) for n = 1..89 (complete sequence)

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

Cf. A225730-A225738.

t = {}; Do[If[n < DivisorSigma[0, n]^(21/10), AppendTo[t, n]], {n, 10^4}]; t

A225738 Number of numbers k such that k < d(k)^(n/10), where d(k) is the number of divisors of k.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 7, 14, 21, 29, 52, 89, 155, 284, 528, 1018, 2046, 4282, 9272, 21466, 50967

Offset: 10

T. D. Noe, May 14 2013

nonn

Cf. A034884 (n < d(n)^2), A056757 (n < d(n)^3), A225729-A225737.

Table[f = 0; Do[If[k < DivisorSigma[0, k]^(n/10), f++], {k, 10^4}]; f, {n, 10, 20}]

A035034 Numbers k such that k >= d(k)^2, where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 5, 7, 9, 11, 13, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81

Offset: 1

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Complement of A034884.

Cf. A000005, A035033, A035035.

Select[Range[100],#>=DivisorSigma[0,#]^2&] (* Harvey P. Dale, May 05 2017 *)

a(n) = n + 52 for n >= 1209. - Amiram Eldar, Aug 29 2020

A056767 Largest number of binary size n (i.e., between (n-1)-th and n-th powers of 2) with the following property: cube of its number of divisors is larger than the number itself.

Original entry on oeis.org

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2046, 4095, 8190, 16380, 32760, 65520, 131040, 262080, 524160, 1048320, 2097144, 4193280, 8386560, 16773900, 33547800, 67095600, 134191200, 268382400, 536215680, 1073709000, 2144142000, 4288284000, 8527559040, 16908091200, 27935107200

Offset: 1

Labos Elemer, Aug 16 2000

			These maximal terms are usually "near" to 2^n. For n=1..10 they are equal to 2^n. At n=21, a(21)=2097144, 1048576 < a(21) < 2097144 = 8*27*7*19*73 has d=128 divisors, of which the cube is d^3d=2097152. So this maximum is near to but still less than d^3.

Cf. A000005, A029837, A034884, A035033, A035034, A035035, A056757-A056767, A056781.

Table[Last@ Select[Range @@ (2^{n - 1, n}), DivisorSigma[0, #]^3 > # &], {n, 22}] (* Michael De Vlieger, Dec 31 2016 *)

a(n) = {k = 2^n; while(numdiv(k)^3 <= k, k--); k;} \\ Michel Marcus, Dec 11 2013

Largest terms of A056757 between 2^(n-1) and 2^n.

a(32) from Michel Marcus, Dec 11 2013

a(33)-a(35) and keyword "full" added by Amiram Eldar, Feb 23 2025

A056781 Prime powers such that the 4th power of the number of divisors is not smaller than the number itself.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 25, 27, 32, 49, 64, 81, 125, 128, 243, 256, 512, 625, 729, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 32768, 65536

Offset: 1

Labos Elemer, Aug 18 2000

For any integers n, d[n]^4>n should form finite albeit very large sequence.

			Equality holds in 12 cases: n=6561=3^8,d[n]=9 and d^4=9^4=3^8=n n=625,d[n]=5, so d^4=n

A000005, A034884, A035033-A035035.

Select[Select[Range[2^16], PrimePowerQ], DivisorSigma[0, #]^4 >= # &] (* Michael De Vlieger, Jul 15 2017 *)

p^w<=(w+1)^4 i.e. p<=(w+1)^(4/w) restricts possible primes and their exponents

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A046525 Numbers common to A033950 and A034884.

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A056757 Cube of number of divisors is larger than the number.

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A035033 Numbers k such that k <= d(k)^2, where d() = number of divisors (A000005).

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A035035 Numbers k such that k > d(k)^2, where d(k) is the number of divisors of k.

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A175495 Positive integers k such that k < 2^d(k), where d(k) is the number of divisors of k.

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A225729 Numbers n such that n < d(n)^(21/10), where d(n) is the number of divisors of n.

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A225738 Number of numbers k such that k < d(k)^(n/10), where d(k) is the number of divisors of k.

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A035034 Numbers k such that k >= d(k)^2, where d(k) is the number of divisors of k.

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A056767 Largest number of binary size n (i.e., between (n-1)-th and n-th powers of 2) with the following property: cube of its number of divisors is larger than the number itself.

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