A285906 -id:A285906 - cp's OEIS frontend

A307870 Numbers k with record values of the ratio d(k)/ud(k) between the number of divisors and the number of unitary divisors.

1, 4, 8, 16, 32, 64, 128, 256, 432, 576, 864, 1296, 1728, 2592, 3456, 5184, 6912, 10368, 15552, 20736, 31104, 41472, 62208, 82944, 93312, 124416, 186624, 248832, 373248, 497664, 746496, 995328, 1119744, 1492992, 2239488, 2592000, 2985984, 3888000, 5184000, 7776000

Offset: 1

Amiram Eldar, May 02 2019

nonn

Numbers k with d(k)/2^omega(k) > d(j)/2^omega(j) for all j < k, where d(k) is the number of divisors of k (A000005), and omega(k) is the number of distinct prime factors of k (A001221), so 2^omega(k) is the number of unitary divisors of k (A034444).
Subsequence of A025487.
The first term that is divisible by the k-th prime is 4, 432, 2592000, 53343360000, 134190022982400000, 35377857659079936000000, 160601747163451186424832000000, 35800939973308629849857487360000000, ...
All the terms are powerful (A001694), since if p is a prime factor of k with multuplicity 1, then k and k/p have the same ratio.

			All squarefree numbers k have d(k)/ud(k) = 1. Thus 4, the first nonsquarefree number, has a record value of d(4)/ud(4) = 3/2 and thus it is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..809

Cf. A000005, A001221, A001694, A025487, A034444, A285906, A307869.

r[n_] := DivisorSigma[0, n]/(2^PrimeNu[n]); rm = 0; n = 1; s = {}; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[s, n]]; n++, {10^7}]; s

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

A335396 Numbers m such that sigma(m)/esigma(m) > sigma(k)/esigma(k) for all k < m, where sigma(m) is the sum of divisors of m (A000203) and esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

1, 2, 6, 30, 96, 210, 480, 1920, 3360, 13440, 36960, 147840, 480480, 1921920, 8168160, 11975040, 32672640, 155675520, 620780160, 1401079680, 2490808320, 2646483840

Offset: 1

Amiram Eldar, Jun 05 2020

			The ratio sigma(m)/esigma(m) for m = 1, 2, ..., 6 is 1, 3/2, 4/3, 7/6, 6/5, 2. The record values occur at m = 1, 2 and 6.

Cf. A000203, A051377, A285906, A335400.

f[n_] := DivisorSigma[1,n]/( Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]); seq = {}; fm = 0; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A335400 Numbers m such that sigma(m)/isigma(m) > sigma(k)/isigma(k) for all k < m, where sigma(m) is the sum of divisors of m (A000203) and isigma(m) is the sum of infinitary divisors of m (A049417).

Original entry on oeis.org

1, 4, 16, 144, 1296, 3600, 20736, 32400, 176400, 518400, 1587600, 12960000, 25401600, 635040000, 3073593600

Offset: 1

Amiram Eldar, Jun 05 2020

			The ratio sigma(m)/isigma(m) for m = 1, 2, 3 and 4 is 1, 1, 1 and 7/5. The record values occur at m = 1 and 4.

Cf. A000203, A049417, A285906, A335396.

fun[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 @@ fun @@@ FactorInteger[n]; f[n_] := DivisorSigma[1,n] / isigma[n]; s={}; fm = 0; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 2 * 10^5}]; s

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A307870 Numbers k with record values of the ratio d(k)/ud(k) between the number of divisors and the number of unitary divisors.

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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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A335396 Numbers m such that sigma(m)/esigma(m) > sigma(k)/esigma(k) for all k < m, where sigma(m) is the sum of divisors of m (A000203) and esigma(m) is the sum of exponential divisors of m (A051377).

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A335400 Numbers m such that sigma(m)/isigma(m) > sigma(k)/isigma(k) for all k < m, where sigma(m) is the sum of divisors of m (A000203) and isigma(m) is the sum of infinitary divisors of m (A049417).

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