A087266 -id:A087266 - cp's OEIS frontend

A087267 a(n) = gcd(n, pi(n)) where pi is A000720.

1, 1, 1, 2, 1, 3, 1, 4, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1, 4, 1, 2, 1, 3, 1, 1, 9, 1, 1, 10, 1, 1, 11, 1, 1, 1, 1, 2, 3, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 5, 3, 1, 1, 2, 1, 8, 1, 2, 1, 1, 1, 2, 9, 2, 1, 6, 1, 1, 1, 1, 1, 4, 1, 1, 3, 1, 7, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 4, 3, 2, 1, 24, 1, 1, 1, 25, 1, 2, 1

Offset: 1

Labos Elemer, Sep 16 2003

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A007053, A000720, A087266, A057809.

Mathematica
```
Table[GCD[w, PrimePi[w]], {w, 1, 256}]
```

a(n) = gcd(n, primepi(n)); \\ Michel Marcus, Apr 22 2018

A087269 Nonprime solutions to gcd(x, pi(x)) = gcd(x, A000720(x)) = 1.

Original entry on oeis.org

1, 9, 12, 18, 21, 25, 26, 28, 32, 34, 35, 36, 42, 45, 49, 52, 55, 57, 60, 65, 68, 69, 70, 74, 76, 81, 84, 85, 86, 87, 88, 91, 95, 98, 99, 104, 106, 110, 111, 112, 119, 121, 128, 129, 130, 133, 135, 141, 143, 145, 147, 155, 158, 159, 160, 161, 162, 165, 170, 172, 177

Offset: 1

Labos Elemer, Sep 16 2003

What is the density of this sequence? - David A. Corneth, Oct 21 2019

			There are 37 primes below the nonprime 162, so pi(162) = 37 and as gcd(162, pi(162)) = gcd(162, 37) = 1, 162 is in the sequence. - _David A. Corneth_, Oct 21 2019

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000720, A007053, A057809, A087266, A087267, A087268.

t=Table[GCD[w, PrimePi[w]], {w, 1, 1000}]; f=Flatten[Position[t, 1]]; cf=Part[f, Flatten[Position[PrimeQ[f], False]]]

first(n) = {n = max(n, 2); my(q = 2, i = 1, t = 1, res = vector(n)); res[1] = 1; forprime(p = 3, oo, for(j = q + 1, p - 1, if(gcd(t, j) == 1, i++; if(i <= n, res[i] = j; , return(res); ) ) ); t++; q = p ) } \\ David A. Corneth, Oct 21 2019

A087268 Solutions to gcd(x, pi(x)) = 1, where pi is A000720.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 12, 13, 17, 18, 19, 21, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 37, 41, 42, 43, 45, 47, 49, 52, 53, 55, 57, 59, 60, 61, 65, 67, 68, 69, 70, 71, 73, 74, 76, 79, 81, 83, 84, 85, 86, 87, 88, 89, 91, 95, 97, 98, 99, 101, 103, 104, 106, 107, 109, 110, 111

Offset: 1

Labos Elemer, Sep 16 2003

nonn

			All primes are included.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A007053, A000720, A087266, A087267, A057809.

Flatten[Position[Table[GCD[w, PrimePi[w]], {w, 1, 256}], 1]]

isok(n) = gcd(n, primepi(n)) == 1; \\ Michel Marcus, Apr 22 2018

A087270 Solutions to gcd(x,pi(x)) = gcd(x, A000720(x)) > 1. Numbers x such that x and pi(x) have common divisor larger than one.

Original entry on oeis.org

4, 6, 8, 10, 14, 15, 16, 20, 22, 24, 27, 30, 33, 38, 39, 40, 44, 46, 48, 50, 51, 54, 56, 58, 62, 63, 64, 66, 72, 75, 77, 78, 80, 82, 90, 92, 93, 94, 96, 100, 102, 105, 108, 114, 115, 116, 117, 118, 120, 122, 123, 124, 125, 126, 132, 134, 136, 138, 140, 142, 144, 146

Offset: 1

Labos Elemer, Sep 16 2003

nonn

Cf. A007053, A000720, A087266-A087269, A057809.

t=Table[GCD[w, PrimePi[w]], {w, 1, 1000}]; Flatten[Position[Sign[t-1], 1]]

A087271 Least number x such that gcd(x, pi(x)) = n.

Original entry on oeis.org

1, 4, 6, 8, 50, 66, 77, 56, 27, 30, 33, 156, 169, 182, 465, 224, 238, 252, 2299, 1380, 189, 902, 207, 96, 100, 1872, 1323, 2464, 1247, 120, 1333, 3168, 528, 1258, 1295, 828, 3441, 2888, 1755, 5800, 1271, 1932, 731, 748, 765, 2852, 2209, 11568, 2695, 4000

Offset: 1

Labos Elemer, Sep 16 2003

nonn

			n=253: a(253)=91586, pi(91586)=8855,
gcd(91586, 8855) = 253 first time.

Cf. A007053, A000720, A087266-A087269, A057809.

f[x_] := GCD[x, PrimePi[x]]; t=Table[0, {257}]; Do[s=f[n]; If[s<258&&t[[s]]==0, t[[s]]=n], {n, 1, 100000}]; t
Module[{tbl=Table[{x,GCD[x,PrimePi[x]]},{x,12000}]},Table[SelectFirst[ tbl,#[[2]]==n&],{n,50}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 12 2020 *)

a(n) = Min{x; gcd(x, A000720(x))=n}.

cp's OEIS Frontend

A087267 a(n) = gcd(n, pi(n)) where pi is A000720.

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A087269 Nonprime solutions to gcd(x, pi(x)) = gcd(x, A000720(x)) = 1.

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A087268 Solutions to gcd(x, pi(x)) = 1, where pi is A000720.

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A087270 Solutions to gcd(x,pi(x)) = gcd(x, A000720(x)) > 1. Numbers x such that x and pi(x) have common divisor larger than one.

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A087271 Least number x such that gcd(x, pi(x)) = n.

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