A292984 -id:A292984 - cp's OEIS frontend

A293185 Bi-unitary highly composite numbers: where the number of bi-unitary divisors of n (A286324) increases to a record.

1, 2, 6, 24, 96, 120, 480, 840, 3360, 7560, 30240, 83160, 272160, 332640, 1081080, 2993760, 4324320, 17297280, 38918880, 69189120, 73513440, 294053760, 661620960, 1176215040, 1396755360, 5587021440, 12570798240, 22348085760, 32125373280, 128501493120

Offset: 1

Amiram Eldar, Oct 01 2017

nonn

Analogous to highly composite numbers (A002182) with number of bi-unitary divisors (A286324) instead of number of divisors (A000005).
The first 12 terms are common with bi-unitary superabundant numbers (A292984).
The record numbers of bi-unitary divisors are 1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 384, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..122
Amiram Eldar, Table of n, a(n), A286324(a(n)) for n = 1..122

Cf. A002182, A286324, A292984.

f[p_, e_] := If[OddQ[e], (e + 1), e]; bdivnum[n_] := If[n == 1, 1, Times @@ (f @@@ FactorInteger[n])]; bm = 0; s = {}; Do[b1 = bdivnum [k]; If[b1 > bm, AppendTo[s, k]; bm = b1], {k, 1, 100000}]; s

a(18)-a(30) from Amiram Eldar, Dec 01 2018

A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994).

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 72, 144, 216, 288, 432, 864, 1728, 2592, 3600, 5400, 7200, 10800, 21600, 43200, 64800, 108000, 129600, 216000, 259200, 324000, 529200, 1058400, 2116800, 3175200, 5292000, 6350400, 10584000, 12700800, 15876000, 31752000, 63504000, 95256000

Offset: 1

Amiram Eldar, May 19 2017

nonn

This sequence is infinite.
a(1) = 1, a(6) = 36, a(15) = 3600 and a(32) = 6350400 are the smallest numbers n such that uphi(n)/phi(n) = 1, 2, 3 and 4. They are squares of 1, 6, 60, and 2520.
Also, coreful superabundant numbers: numbers k with a record value of the coreful abundancy index, A057723(k)/k > A057723(m)/m for all m < k. The two sequences are equivalent since A057723(k)/k = A047994(k)/A000010(k) for all k. - Amiram Eldar, Dec 28 2020

			uphi(k)/phi(k) = 1, 1, 1, 3/2 for k = 1, 2, 3, 4, thus a(1) = 1 and a(2) = 4 since a(4) > a(m) for m < 4.

Amiram Eldar, Table of n, a(n) for n = 1..46

Cf. A000010, A047994, A057723, A285906.

Similar sequences: A002110, A004394, A037992, A061742, A292984, A329882, A333953.

uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@
FactorInteger[n]))[[1]]]; a = {}; rmax = 0; For[k = 0, k < 10^9, k++; r = uphi[k]/EulerPhi[k]; If[r > rmax, rmax = r; a = AppendTo[a, k]]]; a

uphi(n) = my(f = factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1);
lista(nn) = {my(rmax = 0); for (n=1, nn, if ((newr=uphi(n)/eulerphi(n)) > rmax, print1(n, ", "); rmax = newr););} \\ Michel Marcus, May 20 2017

A348273 Noninfinitary superabundant numbers: numbers m such that nisigma(m)/m > nisigma(k)/k for all k < m, where nisigma(m) is the sum of noninfinitary divisors of m (A348271).

Original entry on oeis.org

1, 4, 12, 16, 36, 48, 144, 720, 3600, 25200, 176400, 226800, 1587600, 1940400, 2494800, 17463600, 32432400, 192099600, 227026800, 2497294800, 3632428800, 32464832400, 39956716800

Offset: 1

Amiram Eldar, Oct 09 2021

The least term k with A348271(k)/k > m for m = 1, 2, 3, .... is 36, 3600, 1587600, ...

Cf. A348271.
Subsequence of A348272.
The noninfinitary version of A004394.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742 (exponential), A292984, A329882.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq = {}; rm = -1; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]],{n, 1, 10^6}]; seq

A349111 Powerful superabundant numbers: numbers m such that psigma(m)/m > psigma(k)/k for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 432, 864, 1296, 1728, 2592, 5184, 10368, 15552, 31104, 54000, 108000, 162000, 216000, 324000, 648000, 1296000, 1944000, 3240000, 3888000, 6480000, 9720000, 19440000, 38880000, 58320000, 74088000, 111132000, 222264000, 444528000, 666792000

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

The corresponding record values are 1, 5/4, 13/8, 29/16, 61/32, 125/64, ...

The least term k with psigma(k)/k > m, for m = 2, 3, ..., is 144, 54000, 666792000, ...

Subsequence of A349112.
Cf. A005934, A183097.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742, A292984, A329882, A348273.

f[p_,e_] := (p^(e+1)-1)/(p-1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; rm = 0; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

A361785 Indices of records in the sequence of bi-unitary harmonic means A361782(k)/A361783(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 54, 56, 60, 84, 96, 120, 168, 210, 240, 270, 280, 360, 420, 480, 672, 840, 1080, 1320, 1512, 1680, 1890, 2160, 2310, 2520, 3080, 3360, 4320, 5280, 6048, 7392, 7560, 9240, 10920, 11880, 14040, 15120, 18480, 20790

Offset: 1

Amiram Eldar, Mar 24 2023

nonn

			The harmonic means of the bi-unitary divisors of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. A361782(7)/A361783(7) = 9/5 < 2, and the next record, A361782(8)/A361783(8) = 32/15, occurs at 8. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 8.

Amiram Eldar, Table of n, a(n) for n = 1..192

Cf. A361782, A361783.
Similar sequences: A179971, A348654, A361319.
Other sequences related to records of bi-unitary divisors: A293185, A292983, A292984.

f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; buhmean[1] = 1; buhmean[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{buh, buhmax = 0, s = {}}, Do[buh = buhmean[k]; If[buh > buhmax, buhmax = buh; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[20000]

buhmean(n) = {my(f = factor(n), p, e); n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2)))); }
lista(kmax) = {my(buh, buhmax=0); for(k = 1, kmax, buh = buhmean(k); if(buh > buhmax, buhmax = buh; print1(k, ", "))); }

A348630 Nonexponential superabundant numbers: numbers m such that nesigma(m)/m > nesigma(k)/k for all k < m, where nesigma(m) is the sum of nonexponential divisors of m (A160135).

Original entry on oeis.org

1, 24, 30, 96, 120, 480, 840, 3360, 13440, 30240, 36960, 120960, 147840, 272160, 332640, 1330560, 2993760, 4324320, 17297280, 38918880, 73513440, 220540320, 294053760, 661620960, 1396755360, 2646483840, 5587021440, 12570798240

Offset: 1

Amiram Eldar, Oct 26 2021

The least term k with nesigma(k)/k > m for m = 2, 3, 4, ... is 480, 332640, 1396755360, ...

Cf. A160135, A348604.
Subsequence of A348629.
The nonexponential version of A004394.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742, A292984, A329882, A348273.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; s[1] = 1 ;s[n_] := DivisorSigma[1, n] - esigma[n]; seq = {}; rm = -1; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]],{n, 1, 10^6}]; seq

cp's OEIS Frontend

A293185 Bi-unitary highly composite numbers: where the number of bi-unitary divisors of n (A286324) increases to a record.

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A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994).

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A348273 Noninfinitary superabundant numbers: numbers m such that nisigma(m)/m > nisigma(k)/k for all k < m, where nisigma(m) is the sum of noninfinitary divisors of m (A348271).

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A349111 Powerful superabundant numbers: numbers m such that psigma(m)/m > psigma(k)/k for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

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A361785 Indices of records in the sequence of bi-unitary harmonic means A361782(k)/A361783(k).

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A348630 Nonexponential superabundant numbers: numbers m such that nesigma(m)/m > nesigma(k)/k for all k < m, where nesigma(m) is the sum of nonexponential divisors of m (A160135).

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