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.

A353636 Difference between phi(sigma(n)) and phi(n).

Original entry on oeis.org

0, 1, 0, 4, -2, 2, -2, 4, 6, 2, -6, 8, -6, 2, 0, 22, -10, 18, -10, 4, 4, 2, -14, 8, 10, 0, -2, 12, -20, 16, -14, 20, -4, 2, -8, 60, -18, -2, 0, 8, -28, 20, -22, 4, 0, 2, -30, 44, -6, 40, -8, 18, -34, 14, -16, 8, -4, -4, -42, 32, -30, 2, 12, 94, -24, 28, -34, 4, -12, 24, -46, 72, -36, 0, 20, 12, -28, 24, -46, 28, 56
Offset: 1

Views

Author

Antti Karttunen, May 04 2022

Keywords

Links

Crossrefs

Cf. A000010, A000203, A007431, A062401, A353643, A353647.
Cf. A006872 (positions of zeros), A353637 (their characteristic function).
Cf. A353682 (positions of terms >= 0), A353683 (of terms > 0), A353685 (of terms <= 0), A353686 (of negative terms).
Cf. also A351445.

Programs

  • Mathematica
    a[n_] := EulerPhi[DivisorSigma[1, n]] - EulerPhi[n]; Array[a, 100] (* Amiram Eldar, May 06 2022 *)
  • PARI
    A353636(n) = (eulerphi(sigma(n))-eulerphi(n));

Formula

a(n) = A062401(n) - A000010(n) = A000010(A000203(n)) - A000010(n).
a(n) = Sum_{d|n} (A353647(d) - A007431(d)).

A353644 a(n) = phi(n) / gcd(phi(n), phi(sigma(n))).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 5, 1, 2, 3, 1, 4, 8, 1, 9, 2, 3, 5, 11, 1, 2, 1, 9, 1, 7, 1, 15, 4, 5, 8, 3, 1, 2, 9, 1, 2, 10, 3, 21, 5, 1, 11, 23, 4, 7, 1, 4, 4, 26, 9, 5, 3, 9, 7, 29, 1, 2, 15, 3, 16, 2, 5, 33, 8, 11, 1, 35, 1, 2, 1, 2, 3, 15, 1, 39, 8, 27, 10, 41, 1, 16, 21, 7, 5, 11, 1, 3, 11, 15, 23, 9, 4, 16, 7
Offset: 1

Views

Author

Antti Karttunen, May 06 2022

Keywords

Comments

Denominator of ratio A062401(n) / A000010(n), phi(sigma(n)) / phi(n).

Links

Crossrefs

Cf. A000010, A000203, A062401, A353637, A353643, A353646 (numerators).

Programs

  • Mathematica
    a[n_] := (phi = EulerPhi[n])/GCD[phi, EulerPhi[DivisorSigma[1, n]]]; Array[a, 100] (* Amiram Eldar, May 06 2022 *)
  • PARI
    A353644(n) = { my(ph=eulerphi(n)); (ph / gcd(eulerphi(sigma(n)), ph)); };

Formula

a(n) = A000010(n) / A353643(n) = A000010(n) / gcd(A000010(n), A062401(n)).

A353646 a(n) = phi(sigma(n)) / gcd(phi(n), phi(sigma(n))).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 2, 2, 2, 3, 2, 3, 1, 4, 1, 15, 3, 4, 4, 3, 4, 6, 4, 2, 3, 1, 8, 2, 2, 3, 8, 9, 4, 9, 2, 6, 1, 8, 1, 3, 3, 8, 10, 6, 1, 12, 8, 15, 6, 3, 3, 7, 9, 16, 3, 4, 8, 6, 8, 3, 1, 16, 4, 63, 1, 12, 16, 9, 8, 2, 12, 4, 1, 1, 3, 4, 8, 2, 16, 15, 55, 9, 12, 4, 9, 20, 4, 6, 3, 3, 2, 12, 16, 24, 4, 9, 7, 18, 4, 9
Offset: 1

Views

Author

Antti Karttunen, May 06 2022

Keywords

Comments

Numerator of ratio A062401(n) / A000010(n), phi(sigma(n)) / phi(n).

Links

Crossrefs

Cf. A000010, A000203, A062401, A353636, A353637, A353643, A353644 (denominators).

Programs

  • Mathematica
    a[n_] := (phi = EulerPhi[DivisorSigma[1, n]])/GCD[EulerPhi[n], phi]; Array[a, 100] (* Amiram Eldar, May 06 2022 *)
  • PARI
    A353646(n) = { my(ps=eulerphi(sigma(n))); (ps / gcd(eulerphi(n), ps)); };

Formula

a(n) = A062401(n) / A353643(n) = A062401(n) / gcd(A062401(n), A353636(n)).
Showing 1-3 of 3 results.

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