cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A034884 Numbers k such that k < d(k)^2, where d(k) = A000005(k).

Original entry on oeis.org

2, 3, 4, 6, 8, 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

Views

Author

Erich Friedman

Keywords

Comments

See comment in A175495. - Vladimir Shevelev, May 07 2013
The deficient terms are 2, 3, 4, 8, 10, 14, 15, 16, 32; the first perfect or abundant number not listed is 66 = 2 * 3 * 11; the only term not 7-smooth is 132 = 2^2 * 3 * 11; the largest not divisible by 6 is 140 = 2^2 * 5 * 7. - Peter Munn, Sep 19 2021
The union of this sequence and A276734 has 74 total terms which are all k with floor(sqrt(k)) <= d(k). - Bill McEachen, Apr 07 2023

Crossrefs

Cf. A000005, A035033, A035034, A035035, A033950, A036763, A175495, A276734.

Programs

  • Mathematica
    Select[Range[1300],#Harvey P. Dale, Apr 11 2014 *)
  • PARI
    isok(n) = (n < numdiv(n)^2) \\ Michel Marcus, Jun 07 2013

Extensions

Labos Elemer added the last three terms and observes that this sequence is now complete.

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

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A000005, A034884, A035033, A035034, A035035, A056761, A066693.
Cf. A175495 (k < 2^d(k)), A225729-A225738.

Programs

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

Formula

{ 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

Views

Author

Erich Friedman, Labos Elemer

Keywords

Crossrefs

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

Programs

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Complement of A035033.
Cf. A000005, A034884, A035034.

Programs

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

Formula

a(n) = n + 54 for n >= 1207. - 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

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    Table[Last@ Select[Range @@ (2^{n - 1, n}), DivisorSigma[0, #]^3 > # &], {n, 22}] (* Michael De Vlieger, Dec 31 2016 *)
  • PARI
    a(n) = {k = 2^n; while(numdiv(k)^3 <= k, k--); k;} \\ Michel Marcus, Dec 11 2013

Formula

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

Extensions

a(32) from Michel Marcus, Dec 11 2013
a(33)-a(35) and keyword "full" added by Amiram Eldar, Feb 23 2025

A056766 Smallest term of A056757 (numbers for which the cube of the number of divisors exceeds the number) between 2^(n-1) and 2^n.

Original entry on oeis.org

2, 3, 5, 9, 18, 33, 66, 130, 258, 516, 1026, 2052, 4100, 8200, 16400, 32800, 65550, 131100, 262200, 524400, 1048800, 2097600, 4195200, 8390400, 16783200, 33566400, 67132800, 134265600, 268606800, 537213600, 1074427200, 2148854400, 4297708800, 8627018400, 18897278400
Offset: 1

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

Smallest k so that 2^(n-1) < k <= 2^n and A000005(k)^3 > k.

Examples

			For n=7, 64 < a(7) = 66 < 128, A000005(66)^3 = 8^3 = 512 > 66, and no other such number occurs between 64 and 66.
For n=31, a(31) = 1074427200, 2^30 < a(31) < 2^31; a(31) has 1344 divisors and 1344^3 = 2427715584 > 1074427200. Between 2^30 and a(31) no other numbers occur with this property.

Crossrefs

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

Extensions

a(33)-a(35) from Amiram Eldar, Aug 15 2024
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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