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.

A335053 Odd bi-unitary abundant numbers whose bi-unitary abundancy is closer to 2 than that of any smaller odd bi-unitary abundant number.

Original entry on oeis.org

945, 25515, 46035, 49875, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 396165, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 13351635, 132335385, 159030135, 1756753845, 6561644355, 10394173335, 13455037365, 37456183215
Offset: 1

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Author

Amiram Eldar, May 21 2020

Keywords

Comments

The bi-unitary abundancy of a number k is bsigma(k)/k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Examples

			The bi-unitary abundancies of the first terms are 2.031..., 2.005..., 2.0019..., 2.0018..., 2.0015..., ...

Crossrefs

Cf. A188999, A188263, A293186, A302571, A335052, A335055.

Programs

  • Mathematica
    fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); seq = {}; r = 3; Do[s = bsigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^6, 2}]; seq

A335055 Odd infinitary abundant numbers whose infinitary abundancy is closer to 2 than that of any smaller odd infinitary abundant number.

Original entry on oeis.org

945, 29835, 33345, 43065, 46035, 49875, 83265, 354585, 359205, 361515, 366135, 382305, 389235, 400785, 403095, 407715, 414645, 416955, 423885, 430815, 437745, 442365, 77967015, 132335385, 210124665, 719709375, 724239285, 1756753845, 9665740455, 10394173335
Offset: 1

Views

Author

Amiram Eldar, May 21 2020

Keywords

Comments

The infinitary abundancy of a number k is isigma(k)/k, where isigma(k) is the sum of infinitary divisors of k (A049417).

Examples

			The infinitary abundancies of the first terms are 2.031..., 2.027..., 2.015..., 2.006..., 2.001..., ...

Crossrefs

Cf. A049417, A127666, A188263, A335052, A335053, A335054.

Programs

  • Mathematica
    fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; seq = {}; r = 3; Do[s = isigma[n]/n; If[s > 2 && s < r, AppendTo[seq, n]; r = s], {n, 1, 10^5, 2}]; seq

A336679 Odd exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller odd exponential abundant number.

Original entry on oeis.org

225450225, 385533225, 481583025, 538472025, 672624225, 985646025, 1150227225, 1566972225, 1685513025, 2105433225, 2679615225, 6485886225, 6554064825, 6933060225, 9150077475, 179678493225, 185601564225, 191620685025, 195686793225
Offset: 1

Views

Author

Amiram Eldar, Jul 30 2020

Keywords

Comments

The exponential abundancy of a number k is esigma(k)/k, where esigma(k) is the sum of exponential divisors of k (A051377).
The corresponding values of esigma(k)/k are 2.148..., 2.112..., 2.099..., 2.085..., 2.072..., ...

Crossrefs

The exponential version of A188263.
The odd version of A336254.
Subsequence of A321147.
Similar sequences: A335052, A335053, A335055.
Cf. A051377.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; seq = {}; rm = 3; Do[r = esigma[n]/n; If[r > 2 && r < rm, rm = r; AppendTo[seq, n]], {n, 1, 10^9, 2}]; seq
Showing 1-3 of 3 results.

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

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