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).

A182140 Numbers n such that A060968(n) = A201629(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 255, 257
Offset: 1

Views

Author

José María Grau Ribas, Apr 14 2012

Keywords

Comments

Includes prime numbers and the sequence A071700.
a(n) = A240960(n) for n <= 35. - Reinhard Zumkeller, Aug 05 2014

Links

Crossrefs

Cf. A060968, A201629, A071700, A240960.

Programs

  • Haskell
    a182140 n = a182140_list !! (n-1)
a182140_list = [x | x <- [1..], a060968 x == a201629 x]
-- Reinhard Zumkeller, Aug 05 2014
  • Mathematica
    fa = FactorInteger; A060968[p_, s_] := Which[Mod[p, 4] == 1, p^( s - 1)*(p - 1), Mod[p, 4] == 3, p^(s - 1)*(p + 1), s == 1,2, True, 2^(s + 1)]; A060968[1] = 1; A060968[n_] := Product[A060968[fa[n][[i, 1]], fa[n][[i, 2]]], {i, Length[fa[n]]}]; A201629[n_] := Which[Mod[n, 4] == 1, (n - 1), Mod[n, 4] == 3, (n + 1), True, n]; Select[Range[1000], A060968[#] == A201629[#] &]
  • PARI
    is(n)=my(f=factor(n)[, 1]); n*prod(i=if(n%2, 1, 2), #f, if(f[i]%4==1, 1-1/f[i], 1+1/f[i]))*if(n%4, 1, 2)==if(n%2,(n+1)\4*4,n) \\ Charles R Greathouse IV, Jul 03 2013
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.