cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A250041 Numbers n such that m = floor(n/10) is not coprime to n and, if nonzero, m is also a term of the sequence.

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, 200, 202, 204, 205, 206, 208, 220, 222, 224, 226, 228, 240, 242, 243, 244, 246, 248, 249, 260, 262
Offset: 1

Views

Author

Stanislav Sykora, Dec 07 2014

Keywords

Comments

Equivalent definition 1: Assuming a base b (in this case b=10), let us say that a positive integer k has the property RTNC(b) when m=floor(k/b) is not coprime to k, i.e., gcd(k,m)>1. Then k belongs to this sorted list if (i) it has the property RTNC(b) and (ii) m is either 0 or belongs also to the list.
Equivalent definition 2: Every nonempty prefix of a(n) in base b has the property RTNC(b).
Notes: The acronym RTNC stands for 'Right-Truncated is Not Coprime' (negation of the property RTC defined in A250040). We could also say that a(n) are right-truncatable numbers with property RTNC(b).
This particular list is an infinite subset of A248500.

Examples

			243 is a member because (243,24), (24,2) and (2,0) are noncoprime pairs.
155 is not a member because gcd(15,1)=1.

Links

Crossrefs

Cf. A248500, A250040.
Other lists of right-truncatable numbers with the property RTNC(b): A005823 (b=3), A250037 (b=4), A250039 (b=16), A250043 (b=9), A250045 (b=8), A250047 (b=7), A250049 (b=6), A250051 (b=5).

Programs

  • PARI
    See the link.
  • PARI
    is_rtnc(n, b=10) = {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

Views

Author

Stanislav Sykora, Oct 07 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[ Range@ 120, GCD[#, Floor[#/10]] == 1 &] (* Robert G. Wilson v, Oct 22 2014 *)
  • PARI
    a=vector(20000);
i=n=0;while(i++,if(gcd(i,i\10)==1,a[n++]=i;if(n==#a,break)));a

Formula

{ 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

Views

Author

Stanislav Sykora, Oct 07 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[120],CoprimeQ[#,Floor[#/16]]&] (* Harvey P. Dale, Mar 12 2023 *)
  • PARI
    a=vector(20000);
i=n=0; while(i++, if(gcd(i, i\16)==1, a[n++]=i; if(n==#a, break))); a

Formula

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

Views

Author

Stanislav Sykora, Oct 07 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    Select[Range[150], !CoprimeQ[#, Floor[#/16]] &] (* Amiram Eldar, Nov 30 2024 *)
  • PARI
    a=vector(20000);
i=n=0; while(i++, if(gcd(i, i\16)!=1, a[n++]=i; if(n==#a, break))); a

Formula

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

Views

Author

Stanislav Sykora, Nov 16 2014

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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))

Formula

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.
Showing 1-5 of 5 results.

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