A072155 -id:A072155 - cp's OEIS frontend

A250031 a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))=1.

0, 1, 1, 13, 8, 26, 19, 163, 361, 1223, 1307, 16477, 5749, 83977, 88267, 280817, 147916, 1377406, 2839897, 58552633, 60492571, 63263911, 65468386, 403117367, 549883871, 579629587, 596790577, 1864736021, 1912541636, 29293503812, 59449633388, 969992016739

Offset: 1

Stanislav Sykora, Nov 16 2014

For introduction, see the comments in A250032. The present sequence is obtained when the condition P(m) is identified, for each chosen n>0, with the equality gcd(m,floor(m/n))=1, i.e., P(m)=1 when the equality holds, while P(m)=0 when it does not. Again, the densities d(n) exist and are rational numbers. The value of a(n) is the numerator of d(n), while A250033(n) is the denominator of d(n).

			When n=10, the density of numbers m that are coprime to floor(m/10) turns out to be 1223/2100. Hence a(10) = 1223/2100.
When n=2, all odd numbers qualify, but only the m=2 among even numbers does; hence the density is 1/2 and therefore a(2)=1.
When n=1, only m=1 qualifies, so that the density is 0, and a(1) = 0.

Stanislav Sykora, Table of n, a(n) for n = 1..1000
S. Sykora, On some number densities related to coprimes, Stan's Library, Vol.V, Nov 2014, DOI: 10.3247/SL5Math14.005

Cf. A059956, A248499, A248501, A250032, A250033, A250034.

s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
s(n)=1+s_aux(n,1,1);
a=vector(1000,n,numerator(1-s(n-1)/n))

For n>1, a(n)=A250033(n)-A250032(n), and a(n)/A250033(n)=1-s(n-1)/n, where s(n) A250034(n)/A072155(n).

lim(n->infinity)a(n)/A250033(n) = 1/zeta(2) = A059956.

A071708 Numerator of Sum_{k=1..n} phi(k)/k.

Original entry on oeis.org

1, 3, 13, 8, 52, 19, 163, 361, 1223, 1307, 16477, 5749, 83977, 88267, 280817, 147916, 2754812, 2839897, 58552633, 60492571, 63263911, 65468386, 1612469468, 549883871, 579629587, 596790577, 1864736021, 1912541636, 58587007624, 59449633388, 1939984033478

Offset: 1

N. J. A. Sloane, Jun 28 2002

			1, 3/2, 13/6, 8/3, 52/15, 19/5, 163/35, 361/70, 1223/210, ...

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section I.24, page 27.
Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, 1963.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A072155 (denominators), A000010, A059956.

List([1..35], n-> NumeratorRat( Sum([1..n], k-> Phi(k)/k) ) ); # G. C. Greubel, Aug 25 2019

[Numerator( &+[EulerPhi(k)/k: k in [1..n]] ): n in [1..35]]; // G. C. Greubel, Aug 25 2019

with(numtheory); seq(numer(add(phi(k)/k, k = 1..n)), n = 1..35); # G. C. Greubel, Aug 25 2019

Table[Sum[EulerPhi[k]/k, {k, n}], {n,35}]//Numerator (* G. C. Greubel, Aug 25 2019 *)

a(n) = numerator(sum(k=1,n, eulerphi(k)/k));
vector(35, n, a(n)) \\ G. C. Greubel, Aug 25 2019

[numerator( sum(euler_phi(k)/k for k in (1..n)) ) for n in (1..35)] # G. C. Greubel, Aug 25 2019

Also numerator of Sum_{i=1..n} (mu(i)/i)*floor(n/i). - Ridouane Oudra, Nov 26 2019

a(n)/A072155(n) ~ (6/Pi^2) * n + O(log(n)^(2/3)*log(log(n))^(4/3)). - Amiram Eldar, Sep 18 2022

A250032 a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))>1.

Original entry on oeis.org

1, 1, 1, 11, 7, 19, 16, 117, 269, 877, 1003, 11243, 4261, 56163, 61883, 199663, 107339, 919889, 2009948, 38444267, 41354174, 43432679, 46078049, 266161243, 379669754, 387106183, 407127338, 1258564159, 1322304979, 19229195413, 40830611677, 634491904301, 2638247862269, 2717256540199, 2823435623209, 2886468920107, 1006725304509

Offset: 1

Stanislav Sykora, Nov 16 2014

Let m be any natural number, and P(m) a relational expression on m (i.e., a property of m) evaluating to either 0 (false) or 1 (true). This defines a subset S of natural numbers N for which P(m)=1. When there exists a limit d=limit(M->infinity, Sum(m=1..M, P(m))/M), d is said to be the limit mean density (or just density) of the subset S in N. Now, choose an integer parameter n and set P(m)=gcd(m,floor(m/n))>1. This makes the property P, the corresponding subset S, and the density d all dependent upon n. The reference proves that for any n>0, the density d(n) exists and is a rational number. The value of a(n) is the numerator of d(n), while A250033(n) is the denominator of d(n).

			When n=1, S includes all natural numbers except 1, so d(1)=1. Hence a(1)=1 and A250033(1)=1.
When n=2, S includes all even numbers greater than 2, so d(2)=1/2. Hence a(2)=1 and A250033(2)=2.
When n=10, the subset S is A248500 and d(10)=877/2100. Hence a(10)=877 and A250033(10)=2100.
When n=16, S is A248502 and d(16)=199663/480480. Hence a(16)=199663 and A250033(16)=480480.

Stanislav Sykora, Table of n, a(n) for n = 1..1000
S. Sykora, On some number densities related to coprimes, Stan's Library, Vol. V, Nov 2014, DOI: 10.3247/SL5Math14.005

Cf. A229099, A248500, A248502, A250031, A250033, A250034.

s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
s(n)=1+s_aux(n,1,1);
a=vector(1000,n,numerator(s(n-1)/n))

For n>1, a(n)/A250033(n) = s(n-1)/n, where s(n) = A250034(n)/A072155(n).

lim(n->infinity)a(n)/A250033(n) = 1-1/zeta(2) = A229099.

A250034 Numerators a(n) of the rational-valued function s(n) defined below.

Original entry on oeis.org

1, 3, 11, 7, 38, 16, 117, 269, 877, 1003, 11243, 4261, 56163, 61883, 199663, 107339, 1839778, 2009948, 38444267, 41354174, 43432679, 46078049, 1064644972, 379669754, 387106183, 407127338, 1258564159, 1322304979, 38458390826, 40830611677, 1268983808602

Offset: 1

Stanislav Sykora, Nov 16 2014

a(n) is the numerator (after normalization) of the rational function s(n) = 1-sum(k>0,(-1)^k*sum(p1A072155 (tested up to n=10000). For more information, see also A250031 and A250032.

			n=4: s(4) = 1 - (-1)*(floor(4/2)/2 + floor(4/3)/3)    = 1 + 1 + 1/3  = 7/3, with a(4) = 7 and 3 is indeed A072155(4). - _Wolfdieter Lang_, Dec 02 2014

Stanislav Sykora, Table of n, a(n) for n = 1..1000
S. Sykora, On some number densities related to coprimes, Stan's Library, Vol.V, Nov 2014, DOI: 10.3247/SL5Math14.005

Cf. A250031, A250032, A250033.

s_aux(n,p0,inp)={my(t=0/1,tt=0/1,in=inp,pp);while(1,pp=p0*prime(in);tt=n\pp;if(tt==0,break,t+=tt/pp-s_aux(n,pp,in++)));return(t)};
s(n)=1+s_aux(n,1,1);
a=vector(1000,n,numerator(s(n)))

A333695 Numerators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

Original entry on oeis.org

1, 3, 7, 11, 21, 7, 43, 43, 61, 63, 111, 77, 157, 129, 49, 171, 273, 61, 343, 231, 43, 333, 507, 301, 521, 471, 547, 473, 813, 147, 931, 683, 259, 819, 129, 671, 1333, 1029, 1099, 903, 1641, 43, 1807, 111, 427, 1521, 2163, 399, 2101, 1563, 637, 1727, 2757, 547, 2331

Offset: 1

Ilya Gutkovskiy, Apr 02 2020

			1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

Cf. A000010, A000203, A001157, A018804, A057660, A071708, A072155, A074947, A074949, A333696 (denominators), A346602.

nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 55}] // Numerator
Table[Sum[1/GCD[n, k], {k, n}], {n, 55}] // Numerator
Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 55}] // Numerator

a(n) = numerator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020

a(n) = numerator of Sum_{d|n} phi(n/d) / d.
a(n) = numerator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = numerator of sigma_2(n^2) / (n * sigma_1(n^2)).
a(p) = p^2 - p + 1 where p is prime.
From Amiram Eldar, Nov 21 2022: (Start)
a(n) = numerator(A057660(n)/n).
Sum_{k=1..n} a(k)/A333696(k) ~ c * n^2, where c = zeta(3)/(2*zeta(2)) = 0.365381... (A346602). (End)

A333696 Denominators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 10, 11, 12, 13, 14, 5, 16, 17, 6, 19, 20, 3, 22, 23, 24, 25, 26, 27, 28, 29, 10, 31, 32, 11, 34, 5, 36, 37, 38, 39, 40, 41, 2, 43, 4, 15, 46, 47, 16, 49, 50, 17, 52, 53, 18, 55, 56, 57, 58, 59, 20, 61, 62, 63, 64, 65, 22, 67, 68, 23, 10

Offset: 1

Ilya Gutkovskiy, Apr 02 2020

			1, 3/2, 7/3, 11/4, 21/5, 7/2, 43/7, 43/8, 61/9, 63/10, 111/11, 77/12, 157/13, 129/14, 49/5, ...

Cf. A000010, A000203, A001157, A018804, A057660, A070999, A071000 (fixed points), A071708, A072155, A074947, A074949, A333695 (numerators).

nmax = 70; CoefficientList[Series[Sum[EulerPhi[k] Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[EulerPhi[n/d]/d, {d, Divisors[n]}], {n, 70}] // Denominator
Table[Sum[1/GCD[n, k], {k, n}], {n, 70}] // Denominator
Table[DivisorSigma[2, n^2]/(n DivisorSigma[1, n^2]), {n, 70}] // Denominator

a(n) = denominator(sumdiv(n, d, eulerphi(n/d) / d)); \\ Michel Marcus, Apr 03 2020

a(n) = denominator of Sum_{d|n} phi(n/d) / d.
a(n) = denominator of Sum_{k=1..n} 1 / gcd(n,k).
a(n) = denominator of sigma_2(n^2) / (n * sigma_1(n^2)).

A385561 Numbers m such that (1/m) * Sum_{k=1..m} phi(k)/k is closer to 6/Pi^2 than it is for any number smaller than m, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 16, 22, 28, 36, 66, 96, 100, 126, 156, 190, 330, 430, 540, 820, 876, 1086, 1422, 10596, 10836, 18096, 35796, 55786, 69336, 111100, 168666, 284650, 905950, 1482300, 1745590, 2405560, 2661310, 4023306, 5869956, 17454580, 25670646, 51305346, 79969618, 211025650, 622626790

Offset: 1

Amiram Eldar, Jul 03 2025

nonn

6/Pi^2 is the asymptotic mean of phi(k)/k, i.e., lim_{m->oo} (1/m) * Sum_{k=1..m} phi(k)/k = 6/Pi^2 (Walfisz, 1963; Sándor et al., 2005).

József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 27.
Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Berlin, 1963.

Cf. A000010, A059956, A071708, A072155, A330899, A385562.

seq[lim_] := Module[{s = {}, sum = 0, dm = 1, d}, Do[sum += EulerPhi[k]/k; If[(d = Abs[sum/k - 6/Pi^2]) < dm, dm = d; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]

list(lim) = {my(sm = 0, dm = 1, d); for(k = 1, lim, sm += eulerphi(k)/k; d = abs(sm/k - 6/Pi^2); if(d < dm, dm = d; print1(k, ", ")));}

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A250031 a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))=1.

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A071708 Numerator of Sum_{k=1..n} phi(k)/k.

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A250032 a(n) is the numerator of the density of natural numbers m such that gcd(m,floor(m/n))>1.

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A250034 Numerators a(n) of the rational-valued function s(n) defined below.

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A333695 Numerators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

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A333696 Denominators of coefficients in expansion of Sum_{k>=1} phi(k) * log(1/(1 - x^k)).

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A385561 Numbers m such that (1/m) * Sum_{k=1..m} phi(k)/k is closer to 6/Pi^2 than it is for any number smaller than m, where phi is the Euler totient function (A000010).

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