A227242 -id:A227242 - cp's OEIS frontend

A341749 Numbers k such that gcd(k, phi(k)) > log(log(k)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96

Offset: 1

Amiram Eldar, Feb 18 2021

nonn

First differs from A080197 at n = 28.
Erdős et al. (2008) proved that the asymptotic density of numbers k such that gcd(k, phi(k)) > (log(log(k)))^u for a real number u > 0 is equal to exp(-gamma) * Integral_{t=u..oo} rho(t) dt, where rho(t) is the Dickman-de Bruijn function and gamma is Euler's constant (A001620). For this sequence u = 1, and therefore its asymptotic density is 1 - exp(-gamma) = 0.43854... (A227242).
There are only 8 cyclic numbers (A003277) in this sequence: 1, 2, 3, 5, 7, 11, 13, 15. All the other terms are in A060679. The first term of A060679 which is not in this sequence is 1622.

			16 is a term since gcd(16, phi(16)) = gcd(16, 8) = 8 > log(log(16)) = 1.0197...
17 is not a term since gcd(17, phi(17)) = gcd(17, 16) = 1 < log(log(17)) = 1.0414...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős, Florian Luca and Carl Pomerance, On the proportion of numbers coprime to a given integer, in: J.-M. De Koninck, A. Granville and F. Luca (eds.), Anatomy of Integers, AMS, 2008, pp. 47-64.
Wikipedia, Dickman function.

Cf. A000010, A001620, A003277, A009195, A060679, A080130, A227242, A341750.

Select[Range[100], GCD[#, EulerPhi[#]] > Log[Log[#]] &]

isok(k) = (k==1) || (gcd(k, eulerphi(k)) > log(log(k))); \\ Michel Marcus, Feb 19 2021

A341750 Numbers k such that gcd(k, sigma(k)) > log(log(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 20, 22, 24, 26, 28, 30, 33, 34, 38, 40, 42, 44, 45, 46, 48, 51, 52, 54, 56, 58, 60, 62, 66, 68, 69, 70, 72, 74, 76, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 99, 102, 104, 105, 106, 108, 110, 112

Offset: 1

Amiram Eldar, Feb 18 2021

nonn

Pollack (2011) proved that the asymptotic density of numbers k such that gcd(k, sigma(k)) > (log(log(k)))^u for a real number u > 0 is equal to exp(-gamma) * Integral_{t=u..oo} rho(t) dt, where rho(t) is the Dickman-de Bruijn function and gamma is Euler's constant (A001620). For this sequence u = 1, and therefore its asymptotic density is 1 - exp(-gamma) = 0.43854... (A227242).

There are only 10 terms of A014567 in this sequence: 1, 2, 3, 4, 5, 7, 8, 9, 11, 13.

			15 is a term since gcd(15, sigma(15)) = gcd(15, 24) = 3 > log(log(15)) = 0.996...
16 is not a term since gcd(16, sigma(16)) = gcd(16, 31) = 1 < log(log(16)) = 1.0197...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J., Vol. 60, No. 1 (2011), pp. 199-214.
Wikipedia, Dickman function.

Cf. A000203, A001620, A014567, A080130, A009194, A227242, A341749.

Select[Range[100], GCD[#, DivisorSigma[1, #]] > Log[Log[#]] &]

isok(k) = (k==1) || (gcd(k, sigma(k)) > log(log(k))); \\ Michel Marcus, Feb 20 2021

A351901 Number of permutations of [n] having at least one repeated cycle length.

Original entry on oeis.org

0, 0, 1, 1, 10, 46, 246, 1926, 16080, 143424, 1397520, 16163280, 190902240, 2534113440, 35501044320, 531674569440, 8558324490240, 147103748144640, 2631981703680000, 50393537347829760, 1011054905709004800, 21229069614652569600, 468171587690550374400

Offset: 0

Alois P. Heinz, Feb 24 2022

nonn

			a(2) = 1: (1)(2).
a(3) = 1: (1)(2)(3).
a(4) = 10: (1)(2)(3)(4), (1)(2)(3,4), (1)(2,4)(3), (1)(2,3)(4), (1,4)(2)(3), (1,3)(2)(4), (1,2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Permutation

Cf. A000142, A001620, A007838, A227242.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+b(n-i, min(i-1, n-i))/i))
    end:
a:= n-> n!*(1-b(n$2)):
seq(a(n), n=0..23);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];
a[n_] := n!*(1 - b[n, n]);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 19 2022, after Alois P. Heinz *)

E.g.f.: 1/(1-x) - Product_{j>=1} (1 + x^j/j).
a(n) = A000142(n) - A007838(n).
Limit_{n-> infinity} a(n)/n! = 1 - exp(-gamma) = A227242 = 0.43854... .

A359057 Decimal expansion of 1/(1 - e^(-gamma)).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 14 2022

This constant is mentioned by Andreas Weingartner.

			2.2802910165143604282867468123251090181102824133274380534504187669076628...

Andreas Weingartner, The number of prime factors of integers with dense divisors, arXiv preprint arXiv:2101.11585 [math.NT], 2021.

Cf. A001113, A001620, A080130, A174973, A227242.

RealDigits[1/(1 - Exp[-EulerGamma]), 10, 120][[1]] (* Amiram Eldar, Dec 15 2022 *)

1/(1-exp(-Euler)) \\ Michel Marcus, Dec 15 2022

Equals 1/A227242.
Equals 1/(1 - A080130).
Equals 1/(1 - A001113^(-A001620)).

More terms from Alois P. Heinz, Dec 14 2022

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A341749 Numbers k such that gcd(k, phi(k)) > log(log(k)).

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A341750 Numbers k such that gcd(k, sigma(k)) > log(log(k)).

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A351901 Number of permutations of [n] having at least one repeated cycle length.

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A359057 Decimal expansion of 1/(1 - e^(-gamma)).

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