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 1-4 of 4 results.

A354768 Numbers k such that d(k)/k >= d(m)/m for all m > k, where d(k) is the number-of-divisors function A000005(k).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 18, 24, 30, 36, 48, 60, 72, 84, 90, 120, 144, 180, 240, 252, 360, 420, 480, 504, 540, 720, 840, 900, 1008, 1080, 1260, 1440, 1680, 1800, 2520, 2640, 2880, 3360, 3780, 3960, 5040, 5280, 5400, 5460, 5544, 6300, 7560, 7920, 8400, 10080, 10920, 12600, 15120, 15840, 16380, 18480
Offset: 1

Views

Author

N. J. A. Sloane, Jun 21 2022

Keywords

Comments

Because of the bound d(m) <= 2*sqrt(m), in order for k to be in the sequence it suffices that d(k)/k >= d(m)/m for k < m < (2*k/d(k))^2. - Robert Israel, Jan 23 2023

References

  • David desJardins, Posting to Math Fun Mailing List, Jun 21 2022.

Crossrefs

Cf. A000005, A066523, A354769, A368523.

Programs

  • Maple
    N:= 10^6:
Q:= [seq(numtheory:-tau(k)/k, k=1..N)]:
V:= Vector(10^6):
r:= 2/10^3:
for n from 10^6 to 1 by -1 do
r:= max(Q[n],r);
V[n]:= r;
od:
select(i -> Q[i] >= V[i+1], [$1..10^6-1]); # Robert Israel, Jan 23 2023

Extensions

More terms from Robert Israel, Jan 23 2023

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

Views

Author

Labos Elemer, Jan 09 2002

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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]

A354769 Numbers k such that d(k)/k = sup {d(m)/m | m > k}, where d(k) is the number-of-divisors function A000005(k).

Original entry on oeis.org

1, 8, 18, 90, 8400, 1201200
Offset: 1

Views

Author

N. J. A. Sloane, Jun 22 2022

Keywords

Comments

It is not known if this sequence is infinite.

References

  • David desJardins, Posting to Math Fun Mailing List, Jun 21 2022.

Crossrefs

Cf. A000005, A066523, A354768.

A354770 Numbers k such that d(k)/log(k) sets a new record, where d(k) is the number-of-divisors function A000005(k).

Original entry on oeis.org

2, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 6486480, 7207200, 8648640
Offset: 1

Views

Author

N. J. A. Sloane, Jun 22 2022

Keywords

Comments

A related sequence, not yet in the OEIS, is "Numbers k such that log(d(k))/log(k) > log(d(m))/log(m) for all m > k". It begins 2, 4, 6, 12, 24, 36, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, ..., and up to this point it agrees with A236021 (except that it doesn't include 1). Does it continue to agree with A236021?

Examples

			The values of d(k)/log(k) for k = 2, 3, ... are 2.885390082, 1.820478453, 2.164042562, 1.242669869, 2.232442506, 1.027796685, 1.923593388, 1.365358840, 1.737177928, 0.8340647828, ... and reach record highs at k = 2 (2.885390082...), k = 60 (2.930872040...), and so on.

References

  • David desJardins, Posting to Math Fun Mailing List, Jun 22 2022.

Crossrefs

Cf. A000005, A066523, A236020, A236021, A354768, A354369.

Programs

  • Mathematica
    s = {}; rm = 0; Do[If[(r = DivisorSigma[0, n]/Log[n]) > rm, rm = r; AppendTo[s, n]], {n, 2, 10^5}]; s (* Amiram Eldar, Jun 22 2022 *)

Extensions

More terms from Amiram Eldar, Jun 22 2022
Showing 1-4 of 4 results.

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

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