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 1-5 of 5 results.

A353537 Numbers whose abundancy index is larger than Pi^2/6.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 148, 150
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2022

Keywords

Comments

The abundancy index of a number k is sigma(k)/k, where sigma is the sum of divisors function (A000203).
Pi^2/6 (A013661) is the asymptotic mean of the abundancy indices of the positive integers.
The least odd term is 45 and the least term that is coprime to 6 is 25025.
Davenport (1933) proved that sigma(k)/k possesses a continuous distribution function and that the asymptotic density of numbers with abundancy index that is larger than x exists for all x > 1 and is a continuous function of x. Therefore, this sequence has an asymptotic density.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 4, 41, 436, 4258, 42928, 428557, 4286145, 42864566, 428585795, 4286368677, 42861854540, ... Apparently, the asymptotic density is 0.4286... which means that the distribution of the abundancy indices is skewed with a positive nonparametric skew.

Examples

			4 is a term since sigma(4)/4 = 7/4 = 1.75 > Pi^2/6 = 1.644...

References

  • Harold Davenport, Über numeri abundantes, Sitzungsberichte der Preußischen Akademie der Wissenschaften, phys.-math. Klasse, No. 6 (1933), pp. 830-837.

Links

Crossrefs

Cf. A000203, A005100, A013661, A017665, A017666, A072355, A302991, A330899.
A005101 is a subsequence.
Subsequences: A353538, A353539, A353540, A353541.

Programs

  • Mathematica
    Select[Range[150], DivisorSigma[-1, #] > Pi^2/6 &]
  • PARI
    isok(k) = sigma(k)/k > Pi^2/6; \\ Michel Marcus, Apr 25 2022

A353538 Numbers k such that k and k+1 both have an abundancy index that is larger than Pi^2/6.

Original entry on oeis.org

44, 63, 75, 104, 135, 164, 188, 189, 195, 224, 230, 231, 255, 284, 285, 315, 344, 374, 375, 404, 405, 434, 435, 440, 441, 464, 494, 495, 524, 567, 584, 675, 692, 735, 764, 819, 824, 825, 854, 855, 944, 945, 975, 1034, 1035, 1070, 1071, 1124, 1155, 1196, 1215, 1274
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2022

Keywords

Comments

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 3, 43, 399, 4292, 41306, 413310, 4155925, 414963651, 4149603572, ...
Apparently, the asymptotic density of this sequence is 0.04149...

Examples

			44 is a term since sigma(44)/44 = 84/44 = 1.909... and sigma(45)/45 = 78/45 = 1.733... are both larger than Pi^2/6 = 1.644... .

Links

Crossrefs

Cf. A000203, A013661.
Subsequence of A353537.
Subsequences: A096399, A353539, A353540, A353541.

Programs

  • Mathematica
    q[n_] := DivisorSigma[-1, n] > Pi^2/6; Select[Range[1300], q[#] && q[# + 1] &]
  • PARI
    isok(k) = ((sigma(k)/k) > Pi^2/6) && ((sigma(k+1)/(k+1)) > Pi^2/6); \\ Michel Marcus, Apr 25 2022

A353539 Starts of runs of 3 consecutive numbers whose abundancy index is larger than Pi^2/6.

Original entry on oeis.org

188, 230, 284, 374, 404, 434, 440, 494, 824, 854, 944, 1034, 1070, 1274, 1448, 1664, 1826, 1910, 1988, 2114, 2330, 2534, 2708, 2924, 3014, 3374, 3548, 3590, 3674, 3716, 3794, 3968, 4184, 4388, 4598, 4634, 4850, 5024, 5048, 5228, 5444, 5474, 5564, 5642, 5864, 5894
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2022

Keywords

Comments

The least odd term is a(64391) = 8110375.

Examples

			188 is a term since sigma(188)/188 = 336/188 = 1.787..., sigma(189)/189 = 320/189 = 1.693... and sigma(190)/190 = 360/190 = 1.894... are all larger than Pi^2/6 = 1.644... .

Links

Crossrefs

Cf. A000203, A013661.
Subsequence of A353537 and A353538.
Subsequences: A096536, A353540, A353541.

Programs

  • Mathematica
    q[n_] := DivisorSigma[-1, n] > Pi^2/6; seq[count_, nConsec_] := Module[{tri = q /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ tri, c++; AppendTo[s, k - nConsec]]; tri = Join[Rest[tri], {q[k]}]; k++]; s]; seq[46, 3]
  • PARI
    isok(k) = ((sigma(k)/k) > Pi^2/6) && ((sigma(k+1)/(k+1)) > Pi^2/6) && ((sigma(k+2)/(k+2)) > Pi^2/6); \\ Michel Marcus, Apr 25 2022

A353541 Starts of runs of 5 consecutive numbers whose abundancy index is larger than Pi^2/6.

Original entry on oeis.org

330921744, 4810639922, 5851774772, 7352078822, 7453434624, 9289182174, 13353118622, 13640815824, 14009147072, 14294291372, 14595483174, 14711699274, 17516524022, 17999693072, 21640033412, 21997704572, 22174683684, 24521431022, 24815902472, 25362139074, 25826980022
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2022

Keywords

Examples

			330921744 is a term since sigma(330921744)/330921744 = 2.583..., sigma(330921745)/330921745 = 1.671..., sigma(330921746)/330921746 = 1.647..., sigma(330921747)/330921747 = 1.663... and sigma(330921748)/330921748 = 1.762... are all larger than Pi^2/6 = 1.644... .

Links

Crossrefs

Cf. A000203, A013661.
Subsequence of A353537, A353538, A353539 and A353540.

A353542 a(n) is the least start of a run of n consecutive numbers whose abundancy index is larger than Pi^2/6.

Original entry on oeis.org

4, 44, 188, 8110374, 330921744
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2022

Keywords

Comments

By the Chinese Remainder Theorem, a(n) exists for all n. The proof is similar to the proof that is given for A094268.

Examples

			a(1) = 4 since 4 is the least term of A353537.
a(2) = 44 since 44 is the least term of A353538.

Crossrefs

Cf. A000203, A013661, A094268.
Cf. A353537, A353538, A353539, A353540, A353541.
Showing 1-5 of 5 results.

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