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 1-10 of 19 results. Next

A046526 Numbers common to A033950 and A035033.

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 96, 108, 132, 180, 240, 252, 288, 360, 480, 504, 720, 1260
Offset: 1

Views

Author

Labos Elemer

Keywords

Crossrefs

Cf. A035033, A036763, A046525.

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}.

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

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Complement of A034884.
Cf. A000005, A035033, A035035.

Programs

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

Formula

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

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

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

Views

Author

Labos Elemer, Aug 18 2000

Keywords

Comments

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

Examples

			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

Crossrefs

A000005, A034884, A035033-A035035.

Programs

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

Formula

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

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

Original entry on oeis.org

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

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[1, 10^6 + 1, 2], DivisorSigma[0, #]^3 > # &] (* Michael De Vlieger, Oct 26 2017 *)
  • PARI
    isok(n) = (n % 2) && (numdiv(n)^3 > n); \\ Michel Marcus, Dec 19 2013

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

Views

Author

Labos Elemer, Aug 16 2000

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A035033-A035035, A034884.

Programs

Formula

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

Extensions

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

A046525 Numbers common to A033950 and A034884.

Original entry on oeis.org

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

Views

Author

Labos Elemer

Keywords

Crossrefs

Cf. A035033, A036763, A046526.

Programs

  • Mathematica
    Select[Range[1265], Mod[#, x = DivisorSigma[0, #]] == 0 && # < x^2 &] (* Jayanta Basu, Jun 27 2013 *)
Showing 1-10 of 19 results. Next

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

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