A307870 -id:A307870 - cp's OEIS frontend

A307869 Decimal expansion of the asymptotic mean of d(k)/2^omega(k), where d(k) is the number of divisors of k (A000005) and omega(k) is the number of its distinct prime divisors (A001221).

1, 4, 2, 7, 6, 5, 6, 5, 3, 5, 4, 2, 4, 8, 3, 9, 8, 8, 3, 1, 1, 7, 5, 2, 3, 9, 3, 9, 6, 8, 7, 3, 2, 7, 9, 0, 4, 0, 9, 3, 7, 3, 3, 6, 2, 8, 0, 7, 4, 4, 3, 9, 2, 7, 4, 2, 2, 4, 7, 4, 1, 4, 3, 6, 7, 3, 4, 4, 2, 9, 8, 8, 3, 4, 1, 1, 5, 3, 8, 9, 4, 0, 7, 4, 8, 3, 0, 3, 5, 2, 6, 0, 8, 3, 7, 4, 0, 5, 1, 7, 7, 9, 3, 2, 5

Offset: 1

Amiram Eldar, May 02 2019

Also the asymptotic mean of ratio between the number of divisors and the number of unitary divisors of the integers.

			1.42765653542483988311752393968732790409373362807443...

Ouarda Bouakkaz and Abdallah Derbal, The mean value of the function d(n)/d*(n) in arithmetic progressions, Notes on Number Theory and Discrete Mathematics, Vol. 29, No. 3 (2023), pp. 445-453.
Mihoub Bouderbala, On a sum of a multiplicative function linked to the divisor function over the set of integers B-multiple of 5, Communications in Mathematics, Vol. 31, No. 1 (2023), pp. 369-374; arXiv preprint, arXiv:2212.05549 [math.NT], 2022.
Abdallah Derbal and Meselem Karras, Valeurs moyennes d'une fonction liée aux diviseurs d'un nombre entier, C. R. Acad. Sci. Paris, Ser. I, Vol. 354, No. 6 (2016), pp. 555-558.
Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, p. 79.

Cf. A000005, A001221, A034444, A307870.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{4, -6, 4}, {0, 4, 12}, m]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n/2^n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 + 1/(2 * p * (p-1))) \\ Amiram Eldar, Mar 17 2021

Equals Product_{p prime} 1 + 1/(2 * p * (p-1)).

Equals (Pi^2/6) * Product_{p prime} 1 - 1/(2 * p^2) + 1/(2 * p^3).

More terms from Vaclav Kotesovec, May 29 2020

A308043 Decimal expansion of the asymptotic mean of 2^omega(k)/d(k), where omega(k) is the number of distinct prime divisors of k (A001221) and d(k) is its number of divisors (A000005).

Original entry on oeis.org

8, 1, 9, 1, 9, 0, 9, 6, 4, 1, 4, 8, 9, 9, 1, 9, 0, 8, 1, 8, 0, 3, 6, 5, 6, 6, 0, 3, 8, 1, 3, 7, 3, 5, 8, 2, 7, 2, 2, 2, 6, 8, 8, 5, 2, 4, 7, 9, 7, 1, 8, 4, 9, 8, 5, 8, 2, 1, 4, 6, 6, 0, 3, 7, 6, 2, 1, 1, 7, 4, 3, 5, 0, 4, 7, 2, 2, 0, 4, 0, 2, 2, 0, 8, 7, 0, 7

Offset: 0

Amiram Eldar, May 10 2019

Also the asymptotic mean of the ratio between the number of unitary divisors and the number of divisors of the integers.

			0.81919096414899190818036566038137358272226885247971...

Cf. A000005, A001221, A034444, A307870 (mean of the inverse ratio).

$MaxExtraPrecision = 1000; m = 1000; f[p_] := (1 - 1/p)*(2*p*Log[p/(p - 1)] - 1); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[f[2] * Exp[ NSum[ Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals Product_{p prime} (1-1/p)*(2*p*log(p/(p-1))-1).

A335832 Numbers k with record values of the ratio d(k)/id(k) between the number of divisors and the number of infinitary divisors.

Original entry on oeis.org

1, 4, 16, 144, 256, 1296, 2304, 20736, 518400, 1679616, 5308416, 12960000, 41990400, 132710400, 429981696, 635040000, 1049760000, 3317760000, 10749542400, 31116960000, 51438240000, 162570240000, 268738560000, 2520473760000, 7965941760000, 13168189440000, 167961600000000

Offset: 1

Amiram Eldar, Jun 25 2020

nonn

This sequence is infinite since the ratio d(k)/id(k) is unbounded. For example, for k = 2^(2^m) we have d(k)/id(k) = (2^m+1)/2.

The corresponding record values are 1, 1.5, 2.5, 3.75, 4.5, 6.25, 6.75, 11.25, 16.875, 20.25, ...

Amiram Eldar, Table of n, a(n) for n = 1..96

Subsequence of A025487.

Cf. A000005, A037445, A307870.

id[1] = 1; id[n_] := Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[All, 2]]]; f[1] = 1; f[n_] := DivisorSigma[0, n]/id[n]; seq = {}; fm = 0; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

The ratios d(k)/id(k) for k = 1, 2, 3 and 4 are 1, 1, 1 and 3/2. The record values occur at 1 and 4.

A361937 Numbers k with record values of the ratio A000005(k)/A246600(k) between the total number of divisors of k and the number of divisors d of k such that the bitwise OR of k and d is equal to k.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 336, 420, 840, 1680, 3360, 6720, 7560, 15120, 30240, 60480, 95760, 120960, 176400, 191520, 257040, 352800, 383040, 514080, 1028160, 1681680, 2056320, 2998800, 3112200, 5525520, 5997600, 6224400, 8353800, 12448800, 16216200, 24897600

Offset: 1

Amiram Eldar, Mar 31 2023

This sequence is infinite since the ratio A000005(k)/A246600(k) is unbounded. For example, if k = 2^m then A000005(k)/A246600(k) = m+1.

All the terms except for 1 are in A355670.

			The ratios A000005(k)/A246600(k) for k = 1, 2, 3 and 4 are 1, 2, 1 and 3. The record values, 1, 2 and 3, occur at 1, 2 and 4, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..52

Cf. A000005, A246600, A355670, A359080, A359082, A359083.

Similar sequences: A307870, A335832, A361807.

r[n_] := DivisorSigma[0,n]/DivisorSum[n, Boole[BitOr[#, n] == n] &];
seq[kmax_] := Module[{rm = 0, k = 1, s = {}, r1}, Do[r1 = r[k]; If[r1 > rm, rm = r1; AppendTo[s, k]], {k, 1 , kmax}]; s]; seq[10^6]

r(n) = numdiv(n)/sumdiv(n, d, bitor(d, n) == n);
lista(kmax) = {my(rm = 0, r1); for(k = 1, kmax, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", "))); }

A361807 Numbers k with record values of the ratio A000005(k)/A049419(k) between the number of divisors of k and the number of exponential divisors of k.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 30030, 480480, 510510, 8168160, 9699690, 155195040, 223092870, 3569485920, 6469693230, 103515091680, 200560490130, 3208967842080, 7420738134810, 118731810156960, 304250263527210, 4868004216435360, 13082761331670030, 209324181306720480

Offset: 1

Amiram Eldar, Mar 25 2023

nonn

This sequence is infinite since the ratio A000005(k)/A049419(k) is unbounded. For example, for k = A002110(m) we have A000005(k)/A049419(k) = 2^m.

The corresponding record values are 1, 2, 4, 8, 16, 32, 64, 96, 128, ...

			The ratios A000005(k)/A049419(k) for k = 1, 2, 3, 4, 5 and 6 are 1, 2, 2, 3/2, 2 and 4. The record values, 1, 2 and 4, occur at 1, 2 and 6, the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..84

Subsequence of A025487.
Cf. A000005, A002110, A049419.
Similar sequences: A307870, A335832.

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

r(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 2]+1)/numdiv(f[i, 2])); }
lista(kmax) = {my(rm = 0, r1); for(k = 1, kmax, r1 = r(k); if(r1 > rm, rm = r1; print1(k, ", "))); }

cp's OEIS Frontend

A307869 Decimal expansion of the asymptotic mean of d(k)/2^omega(k), where d(k) is the number of divisors of k (A000005) and omega(k) is the number of its distinct prime divisors (A001221).

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A308043 Decimal expansion of the asymptotic mean of 2^omega(k)/d(k), where omega(k) is the number of distinct prime divisors of k (A001221) and d(k) is its number of divisors (A000005).

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A335832 Numbers k with record values of the ratio d(k)/id(k) between the number of divisors and the number of infinitary divisors.

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A361937 Numbers k with record values of the ratio A000005(k)/A246600(k) between the total number of divisors of k and the number of divisors d of k such that the bitwise OR of k and d is equal to k.

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A361807 Numbers k with record values of the ratio A000005(k)/A049419(k) between the number of divisors of k and the number of exponential divisors of k.

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