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.

A051612 a(n) = sigma(n) - phi(n).

Original entry on oeis.org

0, 2, 2, 5, 2, 10, 2, 11, 7, 14, 2, 24, 2, 18, 16, 23, 2, 33, 2, 34, 20, 26, 2, 52, 11, 30, 22, 44, 2, 64, 2, 47, 28, 38, 24, 79, 2, 42, 32, 74, 2, 84, 2, 64, 54, 50, 2, 108, 15, 73, 40, 74, 2, 102, 32, 96, 44, 62, 2, 152, 2, 66, 68, 95, 36, 124, 2, 94, 52, 120, 2, 171, 2, 78
Offset: 1

Views

Author

Jud McCranie

Keywords

Examples

			a(4) = sigma(4) - phi(4) = 7-2 = 5.

Links

Crossrefs

Cf. A000010, A000203, A110085, A110086, A110087.
Cf. A240960, A292769 (partial sums).

Programs

Formula

a(n) = A000203(n)-A000010(n).
a(n) > 1 for n > 1; a(n) = 2 if and only if n is prime. - Charles R Greathouse IV, May 09 2013
G.f.: Sum_{k>=1} (1 - mu(k))*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Sep 29 2017

A110086 Numbers k such that sigma(k) - phi(k) <= tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 15, 17, 18, 19, 20, 23, 24, 29, 30, 31, 36, 37, 41, 42, 43, 47, 53, 59, 60, 61, 66, 67, 70, 71, 73, 78, 79, 83, 84, 89, 90, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 120, 126, 127, 130, 131, 132, 137, 138, 139, 140, 149, 150, 151
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 11 2005

Keywords

Links

Crossrefs

Union of A000040 and A110085.
Complement of A110086.
Cf. A000005, A000010, A000203, A001221, A051612, A110088, A240960 (subsequence).

Programs

  • Haskell
    a110086 n = a110086_list !! (n-1)
a110086_list = filter (\x -> a051612 x <= a110088 x) [1..]
-- Reinhard Zumkeller, Aug 05 2014
  • Mathematica
    q[k_] := DivisorSigma[1, k] - EulerPhi[k] <= DivisorSigma[0, k]^PrimeNu[k]; Select[Range[160], q] (* Amiram Eldar, Sep 15 2024 *)
  • PARI
    is(n)=sigma(n)-eulerphi(n)<=numdiv(n)^omega(n) \\ Charles R Greathouse IV, Feb 14 2013

Formula

A051612(a(n)) <= A110088(a(n)).

A110088 a(n) = tau(n)^omega(n), where tau = A000005 and omega = A001221.

Original entry on oeis.org

1, 2, 2, 3, 2, 16, 2, 4, 3, 16, 2, 36, 2, 16, 16, 5, 2, 36, 2, 36, 16, 16, 2, 64, 3, 16, 4, 36, 2, 512, 2, 6, 16, 16, 16, 81, 2, 16, 16, 64, 2, 512, 2, 36, 36, 16, 2, 100, 3, 36, 16, 36, 2, 64, 16, 64, 16, 16, 2, 1728, 2, 16, 36, 7, 16, 512, 2, 36, 16, 512, 2, 144, 2, 16, 36, 36, 16
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 11 2005

Keywords

Comments

a(n) depends only on the prime signature of n.

Links

Crossrefs

Cf. A000005, A001221, A110085, A110086, A110087, A240960.

Programs

  • Haskell
    a110088 n = a000005 n ^ a001221 n  -- Reinhard Zumkeller, Aug 05 2014
  • Mathematica
    a[n_] := DivisorSigma[0, n]^PrimeNu[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2024 *)
  • PARI
    a(n) = {my(f = factor(n)); numdiv(f)^omega(f);} \\ Amiram Eldar, Sep 15 2024

Formula

a(n) = A000005(n)^A001221(n).

A353276 a(n) = phi(n) + tau(n)^omega(n) - sigma(n).

Original entry on oeis.org

1, 0, 0, -2, 0, 6, 0, -7, -4, 2, 0, 12, 0, -2, 0, -18, 0, 3, 0, 2, -4, -10, 0, 12, -8, -14, -18, -8, 0, 448, 0, -41, -12, -22, -8, 2, 0, -26, -16, -10, 0, 428, 0, -28, -18, -34, 0, -8, -12, -37, -24, -38, 0, -38, -16, -32, -28, -46, 0, 1576, 0, -50, -32, -88, -20, 388, 0, -58, -36, 392, 0, -27, 0, -62, -48, -68, -20, 368
Offset: 1

Views

Author

Antti Karttunen, Apr 27 2022

Keywords

Links

Crossrefs

Cf. A000005, A000010, A000203, A001221, A051612, A110088.
Cf. A110087 (positions of negative terms), A110086 (of terms >= 0), A110085 (of terms > 0).

Programs

  • Mathematica
    Array[#1 + #3^#2 - #4 & @@ Flatten@ {EulerPhi[#], PrimeNu[#], DivisorSigma[{0, 1}, #]} &, 78] (* Michael De Vlieger, Apr 27 2022 *)
  • PARI
    A353276(n) = (eulerphi(n) + (numdiv(n)^omega(n)) - sigma(n));

Formula

a(n) = A110088(n) - A051612(n) = A000010(n) + A000005(n)^A001221(n) - A000203(n).
a(p) = 0 for all primes p.
Showing 1-4 of 4 results.

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