A248502 -id:A248502 - cp's OEIS frontend

A250039 Numbers n such that m = floor(n/16) is not coprime to n and, if nonzero, m is also a term of the sequence.

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, Dec 07 2014

See the comments in A250041 which all apply, except for the setting of the base, b=16. In particular, they define the property RTNC(b).

This list is an infinite subset of A248502 with which it shares the first 111 entries.

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.
Wikipedia, Coprime integers

Cf. A248502, A250038, A250041.

Other lists of right-truncatable numbers with the property RTNC(b): A005823 (b=3), A250037 (b=4), A250041 (b=10), A250043 (b=9), A250045 (b=8), A250047 (b=7), A250049 (b=6), A250051 (b=5).

PARI
```
See the link.
```

is_rtnc(n, b=16) = {while (((m=gcd(n\b, n)) != 1), if (m == 0, return (1)); n = n\b; ); return (0); } \\ Michel Marcus, Jan 29 2015

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 }.

A248500 Numbers m that are not coprime to A059995(m): floor(m/10).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 24, 26, 28, 30, 33, 36, 39, 40, 42, 44, 46, 48, 50, 55, 60, 62, 63, 64, 66, 68, 69, 70, 77, 80, 82, 84, 86, 88, 90, 93, 96, 99, 100, 102, 104, 105, 106, 108, 110, 120, 122, 123, 124, 126, 128, 129, 130, 140, 142, 144, 146, 147

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 A248499.

			2 is a member because gcd(2,0)=2 > 1.
100 is also a member because gcd(100,10)=10 > 1.
125 is not a member because 125 and 12 are coprime, i.e., gcd(125,12)=1.

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

Cf. A059995, A248499, A248501, A248502.

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

gcd(a(n),floor(a(n)/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.

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.

cp's OEIS Frontend

A250039 Numbers n such that m = floor(n/16) is not coprime to n and, if nonzero, m is also a term of the sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A248500 Numbers m that are not coprime to A059995(m): floor(m/10).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula