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 1-3 of 3 results.

A340087 a(n) = gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 2, 15, 16, 1, 18, 1, 4, 1, 22, 1, 24, 1, 2, 9, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 3, 40, 1, 42, 1, 4, 1, 46, 1, 48, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 9, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 80, 1, 82, 1, 4, 1, 2, 3, 88, 1, 18, 1, 4, 1, 2, 5, 96
Offset: 1

Views

Author

Antti Karttunen, Dec 31 2020

Keywords

Links

Crossrefs

Cf. A091732, A340088, A340089.
Cf. also A049559.

Programs

  • PARI
    ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A340087(n) = gcd(n-1, A091732(n));

Formula

a(n) = gcd(n-1, A091732(n)).
a(n) = A091732(n) / A340088(n).
For n > 1, a(n) = (n-1) / A340089(n).

A340089 a(n) = (n-1) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 5, 21, 1, 23, 1, 25, 13, 3, 1, 29, 1, 31, 8, 33, 17, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 7, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 1, 81, 1, 83, 21, 85, 43, 29, 1, 89
Offset: 1

Views

Author

Antti Karttunen, Dec 31 2020

Keywords

Links

Crossrefs

Cf. A091732, A340087, A340088.
Cf. also A160596.

Programs

  • PARI
    ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A340089(n) = ((n-1)/gcd(n-1, A091732(n)));

Formula

a(n) = (n-1) / A340087(n) = (n-1) / gcd(n-1, A091732(n)).

A345938 a(n) = uphi(n) / gcd(n-1, uphi(n)), where uphi is unitary totient (or unitary phi) function, A047994.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 6, 1, 6, 4, 1, 1, 8, 1, 12, 3, 10, 1, 14, 1, 12, 1, 2, 1, 8, 1, 1, 5, 16, 12, 24, 1, 18, 12, 28, 1, 12, 1, 30, 8, 22, 1, 30, 1, 24, 16, 12, 1, 26, 20, 42, 9, 28, 1, 24, 1, 30, 24, 1, 3, 4, 1, 48, 11, 8, 1, 56, 1, 36, 24, 18, 15, 24, 1, 60, 1, 40, 1, 36, 16, 42, 28, 70, 1, 32, 4, 66, 15
Offset: 1

Views

Author

Antti Karttunen, Jun 29 2021

Keywords

Comments

For all squarefree n (A005117), a(n) = A160595(n), thus if there are any composite solutions to the Lehmer's totient conjecture, then they give also a such a subset of positions of 1's in this sequence that are not powers of primes. See comments in A160595.

Links

Crossrefs

Cf. A005117, A047994, A345937, A345939.
Cf. also A160595, A340088, A345948.

Programs

  • Mathematica
    uphi[1]=1;uphi[n_]:=Times@@(#[[1]]^#[[2]]-1&/@FactorInteger[n]);
a[n_]:=uphi[n]/GCD[n-1,uphi[n]];Array[a,100]  (* Giorgos Kalogeropoulos, Jun 30 2021 *)
  • PARI
    A047994(n) = { my(f=factor(n)~); prod(i=1, #f, (f[1, i]^f[2, i])-1); };
A345938(n) = { my(u=A047994(n)); (u/gcd(n-1, u)); };

Formula

a(n) = A047994(n) / A345937(n) = A047994(n) / gcd(n-1, A047994(n)).
a(2n-1) = A345948(2n-1), for all n >= 1.
Showing 1-3 of 3 results.

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