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 1-7 of 7 results.

A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

901800900, 1542132900, 1926332100, 2153888100, 2690496900, 2822796900, 3942584100, 4487660100, 4600908900, 5127992100, 6267888900, 6742052100, 7162236900, 7305120900, 8421732900, 8969984100, 9866448900, 10203020100, 10718460900, 11723411700, 11787444900, 12528324900
Offset: 1

Views

Author

Amiram Eldar, Oct 04 2019

Keywords

Comments

Aiello et al. found bounds on e-multiperfect numbers, i.e., numbers m such that esigma(m) = k * m for k > 2: 2 * 10^7 for k = 3, and 10^85, 10^320, and 10^1210 for k = 4, 5, and 6. The data of this sequence raise the bound for exponential 3-perfect numbers to 3 * 10^10.
The least odd term is (59#/2)^2 = 924251841031287598942273821762233522616225. The least term which is coprime to 6 is (239#/6)^2 = 3.135... * 10^190.
The least exponential 4-abundant number (esigma(m) >= 4m) is (31#)^2 = 40224510201185827416900. In general, the least exponential k-abundant number (esigma(m) >= k*m), for k > 2, is (A002110(A072986(k)))^2.
The asymptotic density of this sequence is Sum_{n>=1} f(A383699(n)) = 1.325...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, May 06 2025

Links

Crossrefs

Subsequence of A129575.
A383699 is a subsequence.
Cf. A002110, A051377, A072986, A321147.
Cf. A023197, A307112, A285615 (unitary), A293187 (bi-unitary), A300664 (infinitary).

Programs

  • Mathematica
    f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^10], esigma[#] >= 3 # &]

A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 37800, 42336, 43200, 48600, 50400, 54000, 56448, 57600, 59400, 63504, 64800, 70200, 72000, 75600, 79200, 81000, 84672, 86400, 88200, 90000, 91800, 93600, 97200, 98784, 100800, 102600
Offset: 1

Views

Author

Amiram Eldar, Dec 28 2020

Keywords

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
Analogous to A068403 as A308053 is analogous to A005101.
Apparently, the least odd term in this sequence is 3^4 * 5^3 * 7^3 * 11^2 * 13^2 * 17^2 * 19^2 * 23^2 * 29^2 = 3296233276111741840875.
The asymptotic density of this sequence is Sum_{n>=1} f(A364991(n)) = 0.0004006..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). - Amiram Eldar, Aug 15 2023

Examples

			5400 is a term since csigma(5400) = 16380 > 3 * 5400.

Links

Crossrefs

Subsequence of A308053.
Cf. A007947, A057723, A364991 (primitive terms).
Similar sequences: A068403, A285615, A293187, A300664, A307112, A328135.

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] > 3*# &]
  • PARI
    s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) > 3*n; \\ Amiram Eldar, Aug 15 2023

A293187 Bi-unitary 3-abundant numbers: numbers k such that bsigma(k) > 3*k, where bsigma is the sum of the bi-unitary divisors function (A188999).

Original entry on oeis.org

480, 840, 1080, 1320, 1512, 1560, 1680, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3240, 3360, 3480, 3720, 3840, 4320, 4440, 4920, 5160, 5280, 5640, 5880, 6048, 6240, 6360, 6720, 7080, 7320, 7392, 7560, 7680, 8040, 8160, 8520
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2017

Keywords

Comments

Analogous to 3-abundant numbers (A023197) with bi-unitary sigma (A188999) instead of sigma (A000203).

Examples

			480 is in the sequence since bi-unitary sigma(480) = 1512 > 3 * 480.

Links

Crossrefs

Cf. A023197, A285615, A188999, A292982.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] :=  DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bAbundantQ[n_] := bsigma[n] > 3 n; Select[Range[1000], bAbundantQ] (* after Michael De Vlieger at A188999 *)

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.

A336671 Unitary barely 3-abundant: numbers m such that 3 < usigma(m)/m < usigma(k)/k for all numbers k < m, where usigma is the sum of unitary divisors function (A034448).

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 62790, 67830, 79170, 82110, 91770, 103530, 161070, 166530, 709170, 718410, 723030, 732270, 764610, 778470, 801570, 806190, 815430, 829290, 833910, 847770, 861630, 875490, 884730, 155934030, 264670770, 1234205070, 1448478570
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2020

Keywords

Comments

The corresponding values of usigma(m)/m are 3.222..., 3.168...., 3.149..., 3.127..., 3.109..., ...

Links

Crossrefs

The unitary version of A259312.
Subsequence of A285615.
Cf. A034448, A034683, A302570, A336672.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); s = {}; rm = 4; Do[r = usigma[n]/n; If[r > 3 && r < rm, rm = r; AppendTo[s, n]], {n, 1, 10^5}]; s

A336672 Unitary barely 3-deficient numbers: numbers m such that usigma(k)/k < usigma(m)/m < 3 for all numbers k < m, where usigma is the sum of unitary divisors function (A034448).

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 110670, 182910, 898590, 22851570, 26266170, 45255210, 64124970, 265402410, 1374105810, 1631268870
Offset: 1

Views

Author

Amiram Eldar, Jul 29 2020

Keywords

Comments

Unitary 3-deficient numbers are numbers k such that usigma(k) < 3*k, i.e., numbers that are not in A285615.
The corresponding values of usigma(m)/m are 1, 1.5, 2, 2.4, 2.742..., 2.992..., ...
Are terms squarefree? At some point, do we know that a(n) is divisible by primorial(k) for all n > N(k) for some N(k)? - David A. Corneth, Jul 29 2020
Not all the terms are squarefree. E.g., a(12) = 45255210 is divisible by 11^2.

Crossrefs

The unitary version of A307122.
Cf. A034448, A129487, A285615, A302572, A336671.

Programs

  • Mathematica
    usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); s = {}; rm = 0; Do[r = usigma[n]/n; If[r < 3 && r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^5}]; s

A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 79170, 82110, 84630, 85470, 91770, 94710, 99330, 101010, 103530, 108570, 111930, 117390, 122430, 128310, 136290, 140910, 144690, 154770, 161070, 164010, 166530, 168630, 182490, 191730, 205590
Offset: 1

Views

Author

Amiram Eldar, Aug 19 2025

Keywords

Comments

First differs from A067885 at n = 11: A067885(11) = 72930 is not a term of this sequence. a(59) = 510510 is the least term of this sequence that is not in A067885.
Subsequence of A285615 and first differs from it at n = 51: A285615(51) = 390390 is not a term of this sequence.
This sequence is not the same as the sequence of numbers k such that A048250(k) > 3*k which includes all the terms of this sequence but also nonsquarefree numbers, the least of them is 2*A002110(52) = A088860(52) = 2.1248...*10^96.
The least odd term is A002110(17)/2 = 961380175077106319535, the least term that is not divisible by 3 is a(5607800) = 66853496710, and the least term that is coprime to 6 is A002110(52)/6 = 1.7706...*10^95.
If k is a term and m is a squarefree number coprime to k, then k*m is also a term.
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 17, 95, 795, 8162, 86331, 854164, 8372782, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00008... .

Examples

			30030 = 2 * 3 * 5 * 7 * 11 * 13 is a term since it is squarefree, and sigma(30030) = 96768 > 3*30030 = 90090.

Links

Crossrefs

Intersection of A005117 and A068403.
Subsequence of A087248 and A285615.
Cf. A000203, A001221, A002110, A013929, A048250, A088860.

Programs

  • Mathematica
    q[k_] := Module[{f = FactorInteger[k]}, Max[f[[;;, 2]]] == 1 && Times @@ (1 + f[[;; , 1]]) > 3*k]; Select[Range[2*10^5], q]
  • PARI
    isok(k) = {my(f = factor(k)); issquarefree(f) && vecprod(apply(x -> x+1, f[, 1])) > 3*k;}

Formula

A001221(a(n)) >= 6.
Showing 1-7 of 7 results.

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