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.

A129575 Exponential abundant numbers: integers k for which A126164(k) > k, or equivalently for which A051377(k) > 2k.

Original entry on oeis.org

900, 1764, 3600, 4356, 4500, 4900, 6084, 6300, 7056, 8100, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22500, 22932, 25200, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 39600
Offset: 1

Views

Author

Ant King, Apr 28 2007

Keywords

Comments

There are only 52189 exponential abundant numbers less than 50 million, which suggests that these account for approximately 0.1% of all integers.
Includes 36*m for all m coprime to 6 that are not squarefree. - Robert Israel, Feb 19 2019
The asymptotic density of this sequence is Sum_{n>=1} f(A328136(n)) = 0.001043673..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, Sep 02 2022

Examples

			The third integer that is exceeded by its proper exponential divisor sum is 3600. Hence a(3) = 3600.

Links

Crossrefs

Cf. A126164, A051377, A049419, A054979, A054980, A328136.

Programs

  • Maple
    filter:= proc(n)  local L,m,i,j;
  L:= ifactors(n)[2];
  m:= nops(L);
  mul(add(L[i][1]^j, j=numtheory:-divisors(L[i][2])),i=1..m)>2*n
end proc:
select(filter, [$1..10^5]); # Robert Israel, Feb 19 2019
  • Mathematica
    ExponentialDivisors[1]={1};ExponentialDivisors[n_]:=Module[{}, {pr,pows}=Transpose@FactorInteger[n];divpowers=Distribute[Divisors[pows],List];Sort[Times@@(pr^Transpose[divpowers])]];properexponentialdivisorsum[k_]:=Plus@@ExponentialDivisors[k]-k;Select[Range[5 10^4],properexponentialdivisorsum[ # ]># &]
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[40000], esigma[#] > 2*# &] (* Amiram Eldar, May 06 2025 *)
  • PARI
    isok(k) = {my(f = factor(k)); prod(i = 1, #f~, sumdiv(f[i, 2], d, f[i, 1]^d)) > 2*k;} \\ Amiram Eldar, May 06 2025

A321147 Odd exponential abundant numbers: odd numbers k whose sum of exponential divisors A051377(k) > 2*k.

Original entry on oeis.org

225450225, 385533225, 481583025, 538472025, 672624225, 705699225, 985646025, 1121915025, 1150227225, 1281998025, 1566972225, 1685513025, 1790559225, 1826280225, 2105433225, 2242496025, 2466612225, 2550755025, 2679615225, 2930852925, 2946861225, 3132081225
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2018

Keywords

Comments

From Amiram Eldar, Jun 08 2020: (Start)
Exponential abundant numbers that are odd are relatively rare: there are 235290 even exponential abundant number smaller than the first odd term, i.e., a(1) = A129575(235291).
Odd exponential abundant numbers k such that k-1 or k+1 is also exponential abundant number exist (e.g. (73#/5#)^2-1 and (73#/5#)^2 are both exponential abundant numbers, where prime(k)# = A002110(k)). Which pair is the least?
The least exponential abundant number that is coprime to 6 is (31#/3#)^2 = 1117347505588495206025. In general, the least exponential abundant number that is coprime to A002110(k) is (A007708(k+1)#/A002110(k))^2. (End)
The asymptotic density of this sequence is Sum_{n>=1} f(A328136(n)) = 5.29...*10^(-9), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022

Examples

			225450225 is in the sequence since it is odd and A051377(225450225) = 484323840 > 2 * 225450225.

Links

Crossrefs

The exponential version of A005231.
The odd subsequence of A129575.
Cf. A002110, A007708, A051377, A126164, A129485, A293186, A328136.

Programs

  • Mathematica
    esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s={};Do[If[esigma[n]>2n,AppendTo[s,n]],{n,1,10^10,2}]; s

A363169 Powerful abundant numbers: numbers that are both powerful (A001694) and abundant (A005101).

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 784, 800, 864, 900, 968, 972, 1000, 1152, 1296, 1352, 1372, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2500, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000
Offset: 1

Views

Author

Amiram Eldar, May 19 2023

Keywords

Comments

The least odd term is a(90) = 11025, and the least term that is coprime to 6 is 1382511906801025.
Are there two consecutive integers in this sequence? There are none below 10^22.

Examples

			36 = 2^2 * 3^2 is a term since it is powerful, and sigma(36) = 91 > 2*36 = 72.

Links

Crossrefs

Intersection of A001694 and A005101.
Cf. A180114, A363170, A363171.
Subsequences: A307959, A328136, A356871.

Programs

  • Mathematica
    Select[Range[4000], DivisorSigma[-1, #] > 2 && Min[FactorInteger[#][[;;, 2]]] > 1 &]
  • PARI
    is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; }

A356871 Primitive coreful abundant numbers (second definition): coreful abundant numbers (A308053) that are powerful numbers (A001694).

Original entry on oeis.org

72, 108, 144, 200, 216, 288, 324, 400, 432, 576, 648, 784, 800, 864, 900, 972, 1000, 1152, 1296, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000, 4356, 4500, 4608, 4900, 5000, 5184
Offset: 1

Views

Author

Amiram Eldar, Sep 02 2022

Keywords

Comments

For squarefree numbers k, csigma(k) = k, where csigma(k) is the sum of the coreful divisors of k (A057723). Thus, if m is a term (csigma(m) > 2*m) and k is a squarefree number coprime to k, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 2*k*m, so k*m is a coreful abundant number. Therefore, the sequence of coreful abundant numbers (A308053) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful abundant numbers can be calculated from this sequence (see comment in A308053).

Examples

			72 is a term since csigma(72) = 168 > 2 * 72, and 72 = 2^3 * 3^2 is powerful.

Links

Crossrefs

Intersection of A001694 and A308053.
A339940 is a subsequence.
Cf. A057723.
Similar sequences: A307959, A328136.

Programs

  • Mathematica
    f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := If[AllTrue[(fct = FactorInteger[n])[[;;, 2]], #>1 &], Times @@ f @@@ fct, 0]; seq={}; Do[If[s[n] > 2*n, AppendTo[seq, n]], {n, 1, 5000}]; seq

A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

Original entry on oeis.org

900, 1764, 3600, 4356, 4500, 4900, 12348, 47916, 79092, 112500, 605052, 2812500, 13366548, 29647548, 89139564, 231708348, 701538156, 1757812500, 14772192228, 32179382604, 43945312500, 71183762748, 620995547124, 990454107996, 3417547576788, 3488004374652, 10271220141996
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).
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

			The first 6 exponential abundant numbers, 900, 1764, 3600, 4356, 4500 and 4900, have decreasing values of exponential abundancy: 2.4, 2.285..., 2.2, 2.181..., 2.08, 2.057... and therefore they are in this sequence. The next exponential abundant number with a lower exponential abundancy is 12348 with eisgma(12348)/12348 = 2.040...

Links

Crossrefs

The exponential version of A071927.
Subsequence of A001694 and A328136.
Cf. A051377, A057521, A129575, A336253.
Similar sequences: A188263, A302570, A302571, A335054.

Programs

  • Mathematica
    fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 3; 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(23)-a(27) from Amiram Eldar, May 06 2025

A383694 Primitive exponential unitary abundant numbers: the powerful terms of A383693.

Original entry on oeis.org

900, 1764, 4356, 4500, 4900, 6084, 10404, 12348, 12996, 19044, 22500, 30276, 34596, 44100, 47916, 49284, 60516, 66564, 79092, 79524, 86436, 88200, 101124, 108900, 112500, 125316, 132300, 133956, 152100, 161604, 176400, 176868, 181476, 191844, 213444, 217800, 220500
Offset: 1

Views

Author

Amiram Eldar, May 05 2025

Keywords

Comments

First differs from its subsequence A383698 at n = 11.
For squarefree numbers k, eusigma(k) = k, where eusigma is the sum of exponential unitary divisors function (A322857). Thus, if m is a term (eusigma(m) > 2*m) and k is a squarefree number coprime to m, then eusigma(k*m) = eusigma(k) * eusigma(m) = k * eusigma(m) > 2*k*m, so k*m is an exponential unitary abundant number. Therefore, the sequence of exponential unitary abundant numbers (A383693) can be generated from this sequence by multiplying with coprime squarefree numbers.
The least odd term is a(1455) = 225450225, and the least term that is coprime to 6 is 1117347505588495206025.

Examples

			900 is a term since eusigma(900) = 2160 > 2 * 900, and 900 = 2^2 * 3^2 * 5^2 is a powerful number.
6300 is exponential unitary abundant, since eusigma(6300) = 15120 > 2 * 6300, but it is not a powerful number: 6300 = 2^2 * 3^2 * 5^2 * 7. Thus it is not in this sequence. It can be generated as a term of A383693 from a(1) = 900 by 7 * 900 = 6300, since 7 is squarefree and gcd(7, 900) = 1.

Links

Crossrefs

Intersection of A001694 and A383693.
A383698 is a subsequence.
Cf. A005117, A322857, A328136.

Programs

  • Mathematica
    fun[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#] == 1 &]; q[n_] := Min[(f = FactorInteger[n])[[;; , 2]]] > 1 && Times @@ fun @@@ f > 2*n; Select[Range[250000], q]
  • PARI
    fun(p, e) = sumdiv(e, d, if(gcd(d, e/d) == 1, p^d));
isok(k) = {my(f = factor(k)); ispowerful(f) && prod(i = 1, #f~, fun(f[i, 1], f[i, 2])) > 2*k;}

A383699 Primitive exponential 3-abundant numbers: the powerful terms of A328135.

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, May 06 2025

Keywords

Comments

Subsequence of A328135 and first differ from it at n = 25: A328135(25) = 15330615300 is not a term of this sequence.
For squarefree numbers k, esigma(k) = k, where esigma is the sum of exponential divisors function (A051377). Thus, if m is a term (esigma(m) >= 3*m) and k is a squarefree number coprime to m, then esigma(k*m) = esigma(k) * esigma(m) = k * esigma(m) >= 3*k*m, so k*m is an exponential 3-abundant number. Therefore, the sequence of exponential 3-abundant numbers (A328135) can be generated from this sequence by multiplying with coprime squarefree numbers.

Examples

			901800900 is a term since esigma(901800900) = 2905943040 > 3 * 901800900 = 2705402700, and 901800900 = 2^2 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 is a powerful number.

Links

Crossrefs

Intersection of A001694 and A328135.
Subsequence of A328136.
Cf. A051377, A307112.

Programs

  • Mathematica
    pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n];
seq[max_] := Select[pows[max], esigma[#] >= 3 # &]; seq[10^10]
Showing 1-7 of 7 results.

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

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