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 1-7 of 7 results.

A344875 Multiplicative with a(2^e) = 2^(1+e) - 1, and a(p^e) = p^e - 1 for odd primes p.

Original entry on oeis.org

1, 3, 2, 7, 4, 6, 6, 15, 8, 12, 10, 14, 12, 18, 8, 31, 16, 24, 18, 28, 12, 30, 22, 30, 24, 36, 26, 42, 28, 24, 30, 63, 20, 48, 24, 56, 36, 54, 24, 60, 40, 36, 42, 70, 32, 66, 46, 62, 48, 72, 32, 84, 52, 78, 40, 90, 36, 84, 58, 56, 60, 90, 48, 127, 48, 60, 66, 112, 44, 72, 70, 120, 72, 108, 48, 126, 60, 72, 78, 124, 80, 120
Offset: 1

Views

Author

Antti Karttunen, Jun 03 2021

Keywords

Links

Crossrefs

Cf. A047994, A065463, A153151, A344876, A344877, A344878, A344879, A344969, A345947.

Programs

  • Mathematica
    f[2, e_] := 2^(e + 1) - 1; f[p_, e_] := p^e - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 03 2021 *)
  • PARI
    A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
  • Python
    from math import prod
from sympy import factorint
def A344875(n): return prod((p**(1+e) if p == 2 else p**e)-1 for p, e in factorint(n).items()) # Chai Wah Wu, Jun 01 2022

Formula

a(n) = A344878(n) * A344879(n).
Multiplicative with a(p^e) = A153151(p^e). - Antti Karttunen, Jul 01 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (4/5) * A065463 = 0.563553... . - Amiram Eldar, Nov 18 2022

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).

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)).

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

Original entry on oeis.org

0, 0, 0, 0, 0, 3, 0, 0, 0, 8, 0, 6, 0, 11, 3, 0, 0, 16, 0, 13, 6, 19, 0, 15, 0, 24, 0, 35, 0, 9, 0, 0, 9, 32, 10, 48, 0, 35, 12, 45, 0, 16, 0, 38, 23, 43, 0, 30, 0, 48, 15, 45, 0, 51, 30, 42, 18, 56, 0, 41, 0, 59, 21, 0, 23, 49, 0, 96, 21, 52, 0, 57, 0, 72, 24, 70, 39, 60, 0, 60, 0, 80, 0, 36, 30, 83, 27, 118, 0, 61, 59, 131
Offset: 1

Views

Author

Antti Karttunen, Jun 03 2021

Keywords

Comments

Apparently A000961 gives the positions of zeros.

Links

Crossrefs

Cf. A000961, A011772, A344875, A344969, A344973, A344976.
Cf. also A344763, A344765, A344874.

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_] := If[n == 1, 0, A344875[n] - A011772[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)); };
A344876(n) = (A344875(n)-A011772(n));

Formula

a(n) = A344875(n) - A011772(n).
a(n) >= A344976(n).

A344971 a(n) = A344875(n) / gcd(A011772(n), A344875(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 7, 1, 18, 8, 1, 1, 3, 1, 28, 2, 30, 1, 2, 1, 3, 1, 6, 1, 8, 1, 1, 20, 3, 12, 7, 1, 54, 2, 4, 1, 9, 1, 35, 32, 66, 1, 31, 1, 3, 32, 28, 1, 26, 4, 15, 2, 3, 1, 56, 1, 90, 16, 1, 48, 60, 1, 7, 44, 18, 1, 40, 1, 3, 2, 9, 20, 6, 1, 31, 1, 3, 1, 7, 32, 126, 56, 75, 1, 96, 72, 154, 2, 138, 72
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2021

Keywords

Comments

Numerator 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, A344970 (denominators), A344972 (ratio A344875/A011772 floored down), A344973 (and their remainder).

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)); };
A344971(n) = { my(u=A344875(n)); (u/gcd(u, A011772(n))); };

Formula

a(n) = A344875(n) / A344969(n) = A344875(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

A345947 a(n) = gcd(A153151(n), A344875(n)).

Original entry on oeis.org

1, 3, 2, 7, 4, 1, 6, 15, 8, 3, 10, 1, 12, 1, 2, 31, 16, 1, 18, 1, 4, 3, 22, 1, 24, 1, 26, 3, 28, 1, 30, 63, 4, 3, 2, 7, 36, 1, 2, 3, 40, 1, 42, 1, 4, 3, 46, 1, 48, 1, 2, 3, 52, 1, 2, 5, 4, 3, 58, 1, 60, 1, 2, 127, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 80, 3, 82, 1, 4, 1, 2, 3, 88, 1, 18, 7, 4, 3, 2, 1, 96
Offset: 1

Views

Author

Antti Karttunen, Jul 01 2021

Keywords

Links

Crossrefs

Cf. A153151, A344875, A345948, A345949.
Cf. also A344877, A344969, A345937.

Programs

  • Mathematica
    Array[GCD[Which[# < 2, #, IntegerQ[Log2@ #], 2 # - 1, True, # - 1], Times @@ Map[If[#1 == 2, 2^(#2 + 1) - 1, #1^#2 - 1] & @@ # &, FactorInteger[#]]] &, 97] (* Michael De Vlieger, Jul 06 2021 *)
  • PARI
    A153151(n) = if(!n,n,if(!bitand(n,n-1),(n+n-1),(n-1)));
A344875(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^(f[2, i]+(2==f[1, i]))-1)); };
A345947(n) = gcd(A153151(n), A344875(n));

Formula

a(n) = gcd(A153151(n), A344875(n)).
a(n) = A344875(n) / A345948(n).
a(n) = A153151(n) / A345949(n).
a(2n-1) = A345937(2n-1), for n >= 1.
Showing 1-7 of 7 results.

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