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 1-3 of 3 results.

A380929 Numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

Examples

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

Links

Crossrefs

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

A380930 Numbers k such that A380845(k) > 3*k.

Original entry on oeis.org

1080, 2160, 3600, 4320, 7200, 7440, 8640, 11340, 13608, 14400, 14880, 15120, 17280, 18600, 22680, 22860, 27216, 28800, 29760, 30240, 30480, 31752, 33264, 34020, 34560, 37200, 41664, 45360, 45720, 45900, 51408, 53340, 54432, 57600, 59520, 60480, 60960, 61200, 63504
Offset: 1

Views

Author

Amiram Eldar, Feb 08 2025

Keywords

Comments

Analogous to 3-abundant numbers (A068403) with A380845 instead of A000203.

Links

Crossrefs

Cf. A000203, A380845, A380846, A380847.
Subsequence of A068403 and A380929.
Subsequences: A380848, A380931.
Similar sequences: A285615, A293187, A300664, A328135, A340109.

Programs

  • Mathematica
    q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 3*k]; Select[Range[64000], q]
  • PARI
    isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 3*k;}

Formula

1080 is a term since A380845(1080) = 3330 > 3 * 1080 = 3240.

A381070 Numbers k such that A380845(k)/k > A380845(m)/m for all m < k.

Original entry on oeis.org

1, 2, 4, 8, 16, 18, 36, 72, 144, 288, 540, 1080, 2160, 4320, 8640, 17280, 34560, 45360, 68040, 90720, 106680, 136080, 213360, 272160, 320040, 640080, 1280160, 2560320, 2577960, 5155920, 10311840, 15467760, 30935520, 61871040, 123742080, 247484160, 494968320, 681080400
Offset: 1

Views

Author

Amiram Eldar, Feb 13 2025

Keywords

Comments

Analogous to superabundant numbers (A004394) with A380845 instead of A000203.
The least number k for which A380845(k)/k >= 2 is k = a(6) = A380846(1) = 18.
The least number k for which A380845(k)/k >= 3 is k = a(12) = A380930(1) = 1080.
The least number k for which A380845(k)/k >= 4 is k = a(30) = A380931(1) = 5155920.
It seems that A380845(k)/k is unbounded (see the plot in the links section). What is the least number k for which A380845(k)/k >= 5?

Links

Crossrefs

Cf. A000203, A004394, A380845, A380846, A380929, A380930, A380931, A381069.

Programs

  • Mathematica
    r[n_] := Module[{h = DigitCount[n, 2, 1]}, DivisorSum[n, # &, DigitCount[#, 2, 1] == h &]/n]; seq[lim_] := Module[{s = {}, rm = 0, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]
  • PARI
    r(n) = {my(h = hammingweight(n)); sumdiv(n, d, d * (hammingweight(d) == h)) / n;}
list(lim) = {my(rm = 0, r1); for(k = 1, lim, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", "))); }
Showing 1-3 of 3 results.

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