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.

A307122 3-deficient numbers with increasing abundancy: Numbers k such that sigma(m)/m < sigma(k)/k < 3 for all numbers m < k such that sigma(m)/m < 3.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 168, 252, 300, 336, 630, 2268, 2310, 5472, 6804, 20412, 47424, 61236, 161304, 183708, 486096, 551124, 1215216, 1653372, 4081104, 4960116, 14880348, 44641044, 133923132, 401769396, 1205308188, 1631268870, 3615924564, 10847773692
Offset: 1

Views

Author

Amiram Eldar, Mar 26 2019

Keywords

Comments

Analogous to A259312 with 3-deficient numbers instead of 3-abundant numbers.
Analogous to A228450 with ratio 3 instead of 2.
The values of sigma(a(n))/a(n) are 1, 1.5, 1.75, 2, 2.333..., 2.5, 2.527..., 2.583..., 2.8, ...

Crossrefs

Cf. A071927, A023197, A228450, A259312.

Programs

  • Mathematica
    sm=0; seq={}; Do[s=DivisorSigma[1,n]/n; If[s<3 && s>sm, sm=s; AppendTo[seq, n]], {n,1,100000}]; seq

A336253 Exponential barely deficient numbers: exponential deficient numbers whose exponential abundancy is closer to 2 than that of any smaller exponential deficient number.

Original entry on oeis.org

1, 4, 72, 100, 144, 3528, 12100, 15876, 24336, 441000, 1334025, 2205000, 5664400, 24206400, 71267364, 151880976, 3252372552, 9346201200, 13319078472, 26828235000, 347372082000, 1851803856100, 2260121356900, 3198696480100, 5202286387272, 10330374528100, 16316106062400
Offset: 1

Views

Author

Amiram Eldar, Jul 14 2020

Keywords

Comments

The exponential abundancy of a number k is esigma(k)/k, where esigma is the sum of exponential divisors of k (A051377).
Exponential deficient numbers are numbers k with esigma(k)/k < 2. These are numbers that are neither e-perfect (A054979) nor exponential abundant (A129575).
The corresponding values of the exponential abundancy are 1, 1.5, 1.666..., 1.8..., 1.833..., ...
All the terms are powerful numbers (A001694) because esigma(k)/k depends only on the powerful part of k (A057521). - Amiram Eldar, May 06 2025

Examples

			4 is a term since it is exponential deficient, and esigma(4)/4 = 3/2 is higher than esigma(k)/k for all the exponential deficient numbers k < 4.

Links

Crossrefs

Cf. A051377, A054979, A057521, A129575.
Subsequence of A001694.
Similar sequences: A302572, A228450, A262228, A307122, A336252, A336254.

Programs

  • Mathematica
    fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 0; s={}; Do[r = esigma[n]/n; If[r >= 2, Continue[]]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

Extensions

a(21)-a(27) from Amiram Eldar, May 06 2025

A240073 Deficient numbers k for which sigma(k), the sum of divisors of k, reaches a new maximum.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 14, 16, 21, 22, 26, 32, 44, 50, 52, 63, 64, 76, 92, 98, 105, 110, 124, 128, 136, 152, 170, 182, 184, 212, 225, 230, 232, 248, 256, 290, 296, 310, 315, 328, 344, 370, 376, 405, 410, 424, 470, 472, 484, 495, 512, 568, 584, 592, 632, 656
Offset: 1

Views

Author

T. D. Noe, Apr 08 2014

Keywords

Comments

Every power of 2 appears. The deficient number k has sigma(k) < 2*k. In relation to the highly abundant numbers, these numbers might be termed highly deficient numbers.

Links

Crossrefs

Cf. A002093 (highly abundant numbers), A005100 (deficient numbers).
Cf. A228450 (deficient numbers with increasing abundancy).

Programs

  • Mathematica
    t = {}; mn = 0; n = 0; While[Length[t] < 100, n++; d = DivisorSigma[1, n]; If[mn < d < 2*n, AppendTo[t, n]; mn = d]]; t
  • PARI
    lista(kmax) = {my(sigmax = 0, sig); for(k = 1, kmax, sig = sigma(k); if(sig < 2*k && sig > sigmax, sigmax = sig; print1(k, ", ")));} \\ Amiram Eldar, Apr 06 2024

A336252 Infinitary barely deficient numbers: infinitary deficient numbers whose infinitary abundancy is closer to 2 than that of any smaller infinitary deficient number.

Original entry on oeis.org

1, 2, 8, 84, 110, 128, 1155, 3680, 6490, 8200, 8648, 12008, 18632, 32768, 724000, 1495688, 2095208, 3214090, 3477608, 3660008, 5076008, 12026888, 16102808, 26347688, 29322008, 33653888, 73995392, 615206030, 815634435, 2147483648, 42783299288, 80999455688
Offset: 1

Views

Author

Amiram Eldar, Jul 14 2020

Keywords

Comments

The infinitary abundancy of a number k is isigma(k)/k, where isigma is the sum of infinitary divisors of k (A049417).
The corresponding values of the infinitary abundancy are 1, 1.5, 1.875, 1.904..., 1.963..., ...

Examples

			8 is a term since it is infinitary deficient (A129657), and isigma(8)/8 = 15/8 is higher than isigma(k)/k for all the infinitary deficient numbers k < 8.

Crossrefs

Cf. A049417, A129657, A335054.
Similar sequences: A228450, A262228, A302572, A307122, A336253.

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 = 0; Do[s = isigma[n]/n; If[s < 2 && s > r, AppendTo[seq, n]; r = s], {n, 1, 10^6}]; seq

A248816 Numbers that are equal to the arithmetic derivative of the sum of their aliquot parts.

Original entry on oeis.org

152, 284, 4316, 18632, 25484, 2657259, 8394752, 12186976, 17702756, 1172473731, 2147581952, 13716855652, 63831498112
Offset: 1

Views

Author

Paolo P. Lava, Oct 15 2014

Keywords

Comments

Solutions of the equations n = (sigma(n)-n)'.
a(12) > 5*10^9. - Michel Marcus, Nov 01 2014
There could be a relation with terms in A125246 and A228450, since some terms of these sequences are here also. - Michel Marcus, Oct 30 2014
a(14) > 10^11. - Giovanni Resta, May 29 2016

Examples

			Sum of the aliquot parts of 284 is sigma(284) - 284 = 220 and the arithmetic derivative of 220 is 284.~

Crossrefs

Cf. A000203, A003415, A230164.

Programs

  • Maple
    with(numtheory): P:= proc(q) local a,n,p; for n from 1 to q do
a:=(sigma(n)-n)*add(op(2,p)/op(1,p),p=ifactors(sigma(n)-n)[2]);
if n=a then print(n); fi; od; end: P(10^9);
  • PARI
    ad(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);
isok(n) = ad(sigma(n) - n) == n; \\ Michel Marcus, Oct 28 2014

Extensions

a(6)-a(11) from Michel Marcus, Oct 28 2014
a(12)-a(13) from Giovanni Resta, May 29 2016
Showing 1-5 of 5 results.

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