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 1-4 of 4 results.

A344973 a(n) = A344875(n) mod A011772(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 3, 0, 0, 0, 0, 13, 0, 8, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 9, 0, 10, 0, 0, 16, 0, 0, 0, 16, 0, 6, 5, 20, 0, 30, 0, 0, 15, 6, 0, 24, 0, 42, 0, 0, 0, 11, 0, 28, 21, 0, 23, 5, 0, 0, 21, 12, 0, 57, 0, 0, 0, 14, 18, 0, 0, 60, 0, 0, 0, 36, 30, 40, 27, 22, 0, 26, 7, 16, 0, 44, 15, 0, 0, 0, 36
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Links

Crossrefs

Cf. A011772, A344875, A344876, A344969, A344970, A344971, A344972.
Cf. A344974 (positions of zeros).

Programs

  • Mathematica
    b[n_] := If[n == 1, 1, Module[{p, e}, Product[{p, e} = pe;
     If[p == 2, 2^(1 + e) - 1, p^e - 1], {pe, FactorInteger[n]}]]];
c[n_] := Module[{m = 1}, While[Not[IntegerQ[m (m + 1)/(2 n)]], m++]; m];
a[n_] := Mod[b[n], c[n]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)
  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344973(n) = (A344875(n)%A011772(n));

Formula

a(n) = A344875(n) mod A011772(n) = A344876(n) mod A011772(n).

A344969 a(n) = gcd(A011772(n), A344875(n)).

Original entry on oeis.org

1, 3, 2, 7, 4, 3, 6, 15, 8, 4, 10, 2, 12, 1, 1, 31, 16, 8, 18, 1, 6, 1, 22, 15, 24, 12, 26, 7, 28, 3, 30, 63, 1, 16, 2, 8, 36, 1, 12, 15, 40, 4, 42, 2, 1, 1, 46, 2, 48, 24, 1, 3, 52, 3, 10, 6, 18, 28, 58, 1, 60, 1, 3, 127, 1, 1, 66, 16, 1, 4, 70, 3, 72, 36, 24, 14, 3, 12, 78, 4, 80, 40, 82, 12, 2, 1, 1, 2, 88, 1, 1, 1, 30
Offset: 1

Views

Author

Antti Karttunen, Jun 03 2021

Keywords

Links

Crossrefs

Cf. A011772, A344875, A344876, A344970, A344971, A344972, A344973.

Programs

  • Mathematica
    A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];
a[n_] := GCD[A011772[n], A344875[n]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)
  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344969(n) = gcd(A011772(n), A344875(n));

Formula

a(n) = gcd(A011772(n), A344875(n)).
a(n) = gcd(A011772(n), A344876(n)) = gcd(A344875(n), A344876(n)) = gcd(A011772(n), A344973(n)).

A344970 a(n) = A011772(n) / gcd(A011772(n), A344875(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 7, 5, 1, 1, 1, 1, 15, 1, 11, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 11, 1, 7, 1, 1, 19, 1, 1, 1, 5, 1, 16, 9, 23, 1, 16, 1, 1, 17, 13, 1, 9, 1, 8, 1, 1, 1, 15, 1, 31, 9, 1, 25, 11, 1, 1, 23, 5, 1, 21, 1, 1, 1, 4, 7, 1, 1, 16, 1, 1, 1, 4, 17, 43, 29, 16, 1, 35, 13, 23, 1, 47, 19, 1, 1, 1, 11
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Comments

Denominator of the ratio A344875(n)/A011772(n): 1/1, 3/3, 2/2, 7/7, 4/4, 6/3, 6/6, 15/15, 8/8, 12/4, 10/10, 14/8, 12/12, 18/7, 8/5, 31/31, 16/16, 24/8, 18/18, 28/15, 12/6, 30/11, ..., = 1/1, 1/1, 1/1, 1/1, 1/1, 2/1, 1/1, 1/1, 1/1, 3/1, 1/1, 7/4, 1/1, 18/7, 8/5, 1/1, 1/1, 3/1, 1/1, 28/15, 2/1, 30/11, etc.

Links

Crossrefs

Cf. A011772, A344875, A344969, A344971 (numerators), A344972 (ratio floored down), A344974 (positions of ones), A344980 (of terms > 1).

Programs

  • Mathematica
    A011772[n_] := Module[{m = 1}, While[Not[IntegerQ[m(m+1)/(2n)]], m++]; m];
A344875[n_] := Product[{p, e} = pe; If[p == 2, 2^(1+e)-1, p^e-1], {pe, FactorInteger[n]}];
a[n_] := A011772[n]/GCD[A011772[n], A344875[n]];
Array[a, 100] (* Jean-François Alcover, Jun 12 2021 *)
  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344970(n) = { my(u=A011772(n)); (u/gcd(u, A344875(n))); };

Formula

a(n) = A011772(n) / A344969(n) = A011772(n) / gcd(A011772(n), A344875(n)).

A344972 a(n) = floor(A344875(n) / A011772(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 1, 6, 1, 1, 1, 1, 1, 3, 1, 7, 1, 2, 2, 4, 1, 1, 1, 2, 3, 2, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 2, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 7, 1, 3, 1, 1, 1, 3, 2, 2, 2, 6, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 5, 6, 2, 2, 3, 2, 1, 3, 1, 7, 1, 1, 1, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Links

Crossrefs

Cf. A011772, A344875, A344876, A344969, A344970, A344971, A344973.

Programs

  • PARI
    A011772(n) = { if(n==1, return(1)); my(f=factor(if(n%2, n, 2*n)), step=vecmax(vector(#f~, i, f[i, 1]^f[i, 2]))); forstep(m=step, 2*n, step, if(m*(m-1)/2%n==0, return(m-1)); if(m*(m+1)/2%n==0, return(m))); }; \\ From A011772
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A344972(n) = floor(A344875(n)/A011772(n));

Formula

a(n) = (A344875(n)-A344973(n)) / A011772(n) = floor(A344875(n) / A011772(n)).
a(n) = floor(A344971(n) / A344970(n)). - Antti Karttunen, Jun 20 2021
Showing 1-4 of 4 results.

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