A124741 -id:A124741 - cp's OEIS frontend

A057475 Number of k, 1 <= k <= n, such that gcd(n,k) = gcd(n+1,k) = 1.

1, 1, 1, 2, 1, 2, 3, 3, 2, 4, 3, 4, 5, 3, 4, 8, 5, 6, 7, 5, 5, 10, 7, 7, 9, 8, 8, 12, 7, 8, 15, 10, 9, 11, 8, 12, 17, 11, 9, 16, 11, 12, 19, 11, 11, 22, 15, 14, 17, 13, 15, 24, 17, 14, 17, 15, 17, 28, 15, 16, 29, 17, 18, 24, 15, 20, 31, 21, 15, 24, 23, 24, 35, 19, 19, 28, 18, 24, 31, 22

Offset: 1

Leroy Quet, Sep 27 2000

nonn

Number of numbers between 1 and n-1 coprime to n(n+1).

It is conjectured that every positive integer appears. - Jon Perry, Dec 12 2002

			a(8) = 3 because 1, 5 and 7 are all relatively prime to both 8 and 9.
a(9) counts those numbers coprime to 90, i.e., 1 and 7, hence a(9) = 2.

Cf. A000040, A000010, A065474, A118854, A124738, A124739, A124740, A124741.

Cf. A057828, A002378.

[#[k:k in [1..n]| Gcd(n,k) eq Gcd(n+1,k) and Gcd(n,k) eq 1]: n in [1..80]]; // Marius A. Burtea, Oct 15 2019

A057475 := proc(n)
    local a,k ;
    a :=  0;
    for k from 1 to n do
        if igcd(k,n) = 1 and igcd(k,n+1)=1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A057475(n),n=1..80) ; # R. J. Mathar, May 13 2025

a[ n_ ] := Length @ Select[ Range[ n ], GCD[ n, # ] == GCD[ n + 1, # ] == 1 & ]; Table[ a[ n ], {n, 80} ] (* Ray Chandler, Dec 06 2006 *)

newphi(v)=local(vl,fl,np); vl=length(v); np=0; for (s=1,v[1],fl=false; for (r=1,vl,if (gcd(s,v[r])>1,fl=true; break)); if (fl==false,np++)); np
v=vector(2); for (i=1,500,v[1]=i; v[2]=i+1; print1(newphi(v)","))

From Reinhard Zumkeller, May 02 2006: (Start)
a(A000040(n)-1) = A000010(A000040(n)-1);
a(A000040(n)) = A000010(A000040(n)+1)-1;
a(A118854(n)-1) = a(A118854(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 10 2024
a(n) = A057828(A002378(n)). - Ridouane Oudra, May 30 2025

A124738 Irregular table where the n-th row consists of those positive integers which are coprime to both n and n+1 and which are <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 1, 3, 5, 1, 5, 7, 1, 7, 1, 3, 7, 9, 1, 5, 7, 1, 5, 7, 11, 1, 3, 5, 9, 11, 1, 11, 13, 1, 7, 11, 13, 1, 3, 5, 7, 9, 11, 13, 15, 1, 5, 7, 11, 13, 1, 5, 7, 11, 13, 17, 1, 3, 7, 9, 11, 13, 17, 1, 11, 13, 17, 19, 1, 5, 13, 17, 19, 1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 1, 5, 7, 11, 13

Offset: 1

Leroy Quet, Nov 06 2006

The n-th row has A057475(n) terms.

			The positive integers which are coprime to 8 and which are <= 8 are 1,3,5,7. The integers which are coprime to 9 and which are <= 9 are 1, The integers in both these sequences (1,5,7) make up the row of A124738.

Cf. A057475, A124739, A124740, A124741.

f[n_] := Select[Range[n], GCD[n, # ] == GCD[n + 1, # ] == 1 &]; Flatten[Table[f[n], {n, 23}]] (* Ray Chandler, Nov 10 2006 *)

Extended by Ray Chandler, Nov 10 2006

A124740 a(n) = product of those positive integers which are coprime to both n and n+1 and which are <= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 15, 35, 7, 189, 35, 385, 1485, 143, 1001, 2027025, 5005, 85085, 459459, 46189, 20995, 1249937325, 1616615, 7436429, 324342711, 71504125, 132793375, 1452095555625, 7436429, 215656441, 6190283353629375, 75969882625

Offset: 1

Leroy Quet, Nov 06 2006

nonn

			The positive integers which are coprime to 8 and which are <= 8 are 1,3,5,7. The positive integers which are coprime to 9 and which are <= 9 are 1, 2,4,5,7,8. The integers in both these sequences (1,5,7) are multiplied to get a(8) = 35.

Cf. A057475, A124738, A124739, A124741.

f[n_] := Times @@ Select[Range[n], GCD[n, # ] == GCD[n + 1, # ] == 1 &];Table[f[n], {n, 33}] (* Ray Chandler, Nov 10 2006 *)

Extended by Ray Chandler, Nov 10 2006

A124739 a(n) = sum of those positive integers which are coprime to both n and n+1 and which are <= n.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 9, 13, 8, 20, 13, 24, 29, 25, 32, 64, 37, 54, 61, 61, 55, 110, 73, 91, 111, 108, 114, 168, 91, 120, 225, 170, 153, 199, 144, 216, 305, 221, 175, 320, 211, 252, 397, 261, 249, 506, 337, 342, 423, 351, 403, 624, 433, 410, 483, 431, 493, 812, 421, 480

Offset: 1

Leroy Quet, Nov 06 2006

nonn

			The positive integers which are coprime to 8 and which are <= 8 are 1,3,5,7. The integers which are coprime to 9 and which are <= 9 are 1, The integers in both these sequences (1,5,7) are added get a(8) = 13.

Cf. A057475, A124738, A124740, A124741.

f[n_] := Plus @@ Select[Range[n], GCD[n, # ] == GCD[n + 1, # ] == 1 &];Table[f[n], {n, 60}] (* Ray Chandler, Nov 10 2006 *)

Extended by Ray Chandler, Nov 10 2006

cp's OEIS Frontend

A057475 Number of k, 1 <= k <= n, such that gcd(n,k) = gcd(n+1,k) = 1.

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A124738 Irregular table where the n-th row consists of those positive integers which are coprime to both n and n+1 and which are <= n.

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A124740 a(n) = product of those positive integers which are coprime to both n and n+1 and which are <= n.

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A124739 a(n) = sum of those positive integers which are coprime to both n and n+1 and which are <= n.

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Mathematica

Extensions