A250031 -id:A250031 - cp's OEIS frontend

A250033 a(n) gives the denominators for A250031(n) as well as for A250032(n).

1, 2, 2, 24, 15, 45, 35, 280, 630, 2100, 2310, 27720, 10010, 140140, 150150, 480480, 255255, 2297295, 4849845, 96996900, 101846745, 106696590, 111546435, 669278610, 929553625, 966735770, 1003917915, 3123300180, 3234846615, 48522699225, 100280245065, 1604483921040, 6618496174290, 6819056664420, 7019617154550, 7220177644680

Offset: 1

Stanislav Sykora, Nov 16 2014

See the comments in sequences A250031 and A250032.

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.

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,denominator(s(n-1)/n))

a(n)=A250031(n)+A250032(n).

A072155 Denominator of Sum_{k=1..n} phi(k)/k.

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 35, 70, 210, 210, 2310, 770, 10010, 10010, 30030, 15015, 255255, 255255, 4849845, 4849845, 4849845, 4849845, 111546435, 37182145, 37182145, 37182145, 111546435, 111546435, 3234846615, 3234846615, 100280245065, 200560490130, 200560490130

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, ...

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

Cf. A071708 (numerators), A250031, A250032, A250034.

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

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

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

Table[Sum[EulerPhi[k]/k,{k,n}],{n,40}]//Denominator (* Harvey P. Dale, Jun 08 2017 *)

a(n) = denominator(sum(k=1, n, eulerphi(k)/k)); \\ Michel Marcus, Jan 26 2015

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

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

A248499 Numbers m that are coprime to A059995(m): floor(m/10).

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 31, 32, 34, 35, 37, 38, 41, 43, 45, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 65, 67, 71, 72, 73, 74, 75, 76, 78, 79, 81, 83, 85, 87, 89, 91, 92, 94, 95, 97, 98, 101, 103, 107, 109, 111, 112

Offset: 1

Stanislav Sykora, Oct 07 2014

Definition of "being coprime" and special-case conventions are as in Wikipedia. In particular, when m < 10 then floor(m/10) = 0, and zero is coprime only to 1. The complementary sequence is A248500. Note: The first 57 terms a(n) coincide with A069715, but the two sequences are different.

The limit mean density of these numbers exists and equals 1223/2100 = A250031(10)/A250033(10). - Stanislav Sykora, Dec 08 2014

			1 is a term because gcd(1,0) = 1.
123 is not a term because gcd(123,12) = 3.
165 is a term because 165 and 16 are coprime.

Stanislav Sykora, Table of n, a(n) for n = 1..20000
Stanislav Sykora, On some number densities related to coprimes, Stan's Library, Vol. V, Nov 2014, DOI: 10.3247/SL5Math14.005.
Wikipedia, Coprime integers.

Cf. A059995, A248500, A248501, A248502, A250031, A250033.

Select[ Range@ 120, GCD[#, Floor[#/10]] == 1 &] (* Robert G. Wilson v, Oct 22 2014 *)

a=vector(20000);
i=n=0;while(i++,if(gcd(i,i\10)==1,a[n++]=i;if(n==#a,break)));a

{ m : gcd(m, floor(m/10)) = 1 }.

A248501 Numbers m that are coprime to floor(m/16).

Original entry on oeis.org

1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 97, 101, 103, 107, 109, 113, 114, 115

Offset: 1

Stanislav Sykora, Oct 07 2014

Definition of 'being coprime' and special-case conventions are as in Wikipedia. In particular, when m < 16 then floor(m/16) = 0, and zero is coprime only to 1. The complementary sequence is A248502.

The asymptotic density of this sequence is A250031(16)/A250033(16) = 280817/480480 = 0.58445... . - Amiram Eldar, Nov 30 2024

			1 is a term because gcd(1,0) = 1.
2 is not a term because gcd(2,0) = 2.
129 is a term because 129 is coprime to floor(129/16) = 8.

Stanislav Sykora, Table of n, a(n) for n = 1..20000
Wikipedia, Coprime integers.

Cf. A248499, A248500, A248502, A250031, A250033.

Select[Range[120],CoprimeQ[#,Floor[#/16]]&] (* Harvey P. Dale, Mar 12 2023 *)

a=vector(20000);
i=n=0; while(i++, if(gcd(i, i\16)==1, a[n++]=i; if(n==#a, break))); a

gcd(a(n),floor(a(n)/16)) = 1.

A248502 Numbers m that are not coprime to floor(m/16).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 34, 36, 38, 40, 42, 44, 46, 48, 51, 54, 57, 60, 63, 64, 66, 68, 70, 72, 74, 76, 78, 80, 85, 90, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 119, 126, 128, 130, 132, 134, 136, 138, 140, 142

Offset: 1

Stanislav Sykora, Oct 07 2014

Definition of 'being coprime' and special-case conventions are as in Wikipedia. In particular, when m < 16 then floor(m/16) = 0, and zero is coprime only to 1. The complementary sequence is A248501.

The asymptotic density of this sequence is 1 - A250031(16)/A250033(16) = 199663/480480 = 0.415549... . - Amiram Eldar, Nov 30 2024

			2 is a term because gcd(2,0) = 2 > 1.
21 is not a term because floor(21/16) = 1 and 1 is coprime to any number.
200 is a term because floor(200/16) = 12 and gcd(200,12) = 4 > 1.

Stanislav Sykora, Table of n, a(n) for n = 1..20000
Wikipedia, Coprime integers.

Cf. A248499, A248500, A248501, A250031, A250033.

Select[Range[150], !CoprimeQ[#, Floor[#/16]] &] (* Amiram Eldar, Nov 30 2024 *)

a=vector(20000);
i=n=0; while(i++, if(gcd(i, i\16)!=1, a[n++]=i; if(n==#a, break))); a

gcd(a(n),floor(a(n)/16)) > 1.

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)))

cp's OEIS Frontend

A250033 a(n) gives the denominators for A250031(n) as well as for A250032(n).

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A072155 Denominator of Sum_{k=1..n} phi(k)/k.

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A248499 Numbers m that are coprime to A059995(m): floor(m/10).

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A248501 Numbers m that are coprime to floor(m/16).

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A248502 Numbers m that are not coprime to floor(m/16).

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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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