A248499 -id:A248499 - cp's OEIS frontend

A250040 Numbers n such that m = floor(n/10) is coprime to n and, if nonzero, m is also a term of the sequence.

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 101, 103, 107, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 125, 127, 131, 132, 133, 134, 135, 136, 137, 138, 139, 141, 143, 145, 149, 151, 152, 154, 157, 158, 161, 163, 165, 167, 169, 171, 172, 173, 174, 175

Offset: 1

Stanislav Sykora, Dec 07 2014

Equivalent definition 1: Assuming a base b (in this case b=10), let us say that a positive integer k has the property RTC(b) when m=floor(k/b) is coprime to k, i.e., gcd(k,m)=1. Then k belongs to this sorted list if (i) it has the property RTC(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 RTC(b).
Notes: The acronym RTC stands for 'Right-Truncated is Coprime'. We could also say that a(n) are right-truncatable numbers with property RTC(b).
This particular list is an infinite subset of A248499.

			149, 14, and 1 are members because (149,14), (14,1) and (1,0) are all coprime pairs.
67 is not a member because gcd(67,7)=1, but gcd(6,0)=6.

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. A248499, A250041.

Other lists of right-truncatable numbers with the property RTC(b): A250036 (b=4), A250038 (b=16), A250042 (b=9), A250044 (b=8), A250046 (b=7), A250048 (b=6), A250050 (b=5).

F:= proc(a) seq(10*a+d, d = select(t -> igcd(a,t)=1, [$0..9])) end proc:
B[1]:= [1]:
for i from 2 to 4 do
  B[i]:= map(F,B[i-1]);
od:
ListTools:-Flatten([seq(B[i],i=1..4)]); # Robert Israel, Jan 04 2015

PARI
```
See the link.
```

is_rtc(n, b=10) = {while (((m=gcd(n\b, n)) == 1), if (m == 0, return (1)); if ((n=n\b) == 0, return (1));); return (0);} \\ Michel Marcus, Jan 17 2015

A250031 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

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.

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.

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.

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A250040 Numbers n such that m = floor(n/10) is coprime to n and, if nonzero, m is also a term of the sequence.

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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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A248500 Numbers m that are not 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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