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.

Previous Showing 21-26 of 26 results.

A127474 Triangle, square of A054522.

Original entry on oeis.org

1, 2, 1, 3, 0, 4, 4, 3, 0, 4, 5, 0, 0, 0, 16, 6, 3, 8, 0, 0, 4, 7, 0, 0, 0, 0, 0, 36, 8, 7, 0, 12, 0, 0, 0, 16, 9, 0, 16, 0, 0, 0, 0, 0, 36, 10, 5, 0, 0, 32, 0, 0, 0, 0, 16
Offset: 1

Views

Author

Gary W. Adamson, Jan 15 2007

Keywords

Comments

Right border = A127473, squares of phi(n) terms. Row sums = A057660: (1, 3, 7, 11, 21, ...).

Examples

			First few rows of the triangle:
  1;
  2, 1;
  3, 0, 4;
  4, 3, 0,  4;
  5, 0, 0,  0, 16;
  6, 3, 8,  0,  0, 4;
  7, 0, 0,  0,  0, 0, 36;
  8, 7, 0, 12,  0, 0,  0, 16;
  ...

Crossrefs

Cf. A127473, A057660, A054522.

Formula

(A054522)^2 as an infinite lower triangular matrix.

A362414 a(n) = gcd(n, phi(n)^2) / gcd(n, phi(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 3, 5
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 19 2023

Keywords

Comments

a(n) = 1 if n is squarefree.

Links

Crossrefs

Cf. A000010, A000188, A009195.

Programs

  • Magma
    [Gcd(n, EulerPhi(n)^2) / Gcd(n, EulerPhi(n)): n in [1..100]];
  • Mathematica
    A362414[n_]:=With[{p=EulerPhi[n]},GCD[n,p^2]/GCD[n,p]];
Array[A362414,100] (* Paolo Xausa, Oct 22 2023 *)
  • PARI
    a(n)=my(f=eulerphi(n)); gcd(n,f^2)/gcd(n,f) \\ Charles R Greathouse IV, May 03 2023

Formula

a(n) = gcd(n,A127473(n)) / A009195(n).
1 <= a(n) <= sqrt(n). The lower bound is sharp (squarefree numbers), as is the upper bound (squares of primes). - Charles R Greathouse IV, May 03 2023

A113793 Triangle read by rows: T(n,m) = phi(n - m + 1) * phi(m), n >= 1, m >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 4, 4, 4, 2, 6, 2, 8, 4, 8, 2, 6, 4, 6, 4, 8, 8, 4, 6, 4, 6, 4, 12, 4, 16, 4, 12, 4, 6, 4, 6, 8, 12, 8, 8, 12, 8, 6, 4, 10, 4, 12, 8, 24, 4, 24, 8, 12, 4, 10, 4, 10, 8, 12, 16, 12, 12, 16, 12, 8, 10, 4, 12, 4, 20, 8, 24, 8, 36, 8, 24, 8, 20, 4, 12
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Aug 25 2008

Keywords

Examples

			{1},
{1, 1},
{2, 1, 2},
{2, 2, 2, 2},
{4, 2, 4, 2, 4},
{2, 4, 4, 4, 4, 2},
{6, 2, 8, 4, 8, 2, 6},
{4, 6, 4, 8, 8, 4, 6, 4},
{6, 4, 12, 4, 16, 4, 12, 4, 6},
{4, 6, 8, 12, 8, 8, 12, 8, 6, 4},
{10, 4, 12, 8, 24, 4, 24, 8, 12, 4, 10}

Crossrefs

Column 1 and leading diagonal give A000010.
Middle diagonal gives A127473.
Row sums give A065093.

Programs

  • Mathematica
    T[n_, m_] = EulerPhi[n - m + 1]*EulerPhi[m + 1]; Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Formula

T(n,m) = A000010(m)*A000010(n-m+1), n >= 1, m >= 1. - Omar E. Pol, Jan 14 2025

Extensions

Name corrected and more terms added by Omar E. Pol, Jan 14 2025

A117295 a(n) = phi(n)*(n - phi(n)).

Original entry on oeis.org

0, 1, 2, 4, 4, 8, 6, 16, 18, 24, 10, 32, 12, 48, 56, 64, 16, 72, 18, 96, 108, 120, 22, 128, 100, 168, 162, 192, 28, 176, 30, 256, 260, 288, 264, 288, 36, 360, 360, 384, 40, 360, 42, 480, 504, 528, 46, 512, 294, 600, 608, 672, 52, 648, 600, 768, 756, 840, 58, 704, 60
Offset: 1

Views

Author

Luc Stevens (lms022(AT)yahoo.com), Apr 23 2006

Keywords

Examples

			a(8) = phi(8)*(8 - phi(8)) = 4*4 = 16.

Links

Crossrefs

Cf. A000010, A002618, A051953, A127473.
Cf. A059956, A065464.

Programs

  • Mathematica
    a[n_] := Module[{phi = EulerPhi[n]}, phi*(n - phi)]; Array[a, 100] (* Amiram Eldar, Dec 21 2023 *)
  • PARI
    a(n) = eulerphi(n)*(n-eulerphi(n));

Formula

For n > 1, a(n) = Sum_{k=1..n-1} PHI(k,n)^2 where PHI(k,n) = phi(n)*mu(n/GCD(k,n))/phi(n/GCD(k,n)), and has been considered by C. Nicol under the name G(n). - Michel Marcus, Nov 11 2012
From Amiram Eldar, Dec 21 2023: (Start)
a(n) = A002618(n) - A127473(n).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = A059956 - A065464 = 0.179677... . (End)

Extensions

Offset corrected by Georg Fischer, Mar 17 2023

A347293 Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(1 + (i-1) * (k-1),n) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 4, 8, 4, 8, 5, 9, 9, 9, 9, 6, 15, 10, 9, 10, 15, 7, 13, 13, 13, 13, 13, 13, 8, 20, 8, 20, 8, 20, 8, 20, 9, 21, 21, 9, 21, 21, 9, 21, 21, 10, 27, 18, 27, 18, 15, 18, 27, 18, 27, 11, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 12, 40, 20, 24, 20, 40, 12, 40, 20, 24, 20, 40
Offset: 1

Views

Author

Werner Schulte, Jan 23 2022

Keywords

Comments

Triangle without column 1 is symmetrical.
Conjecture: Let f be an arbitrary arithmetic function. Define for n > 0 the sequence a(f; n) = Sum_{i=1..n, k=1..n} f(gcd(1 + (i-1) * (k-1),n)); then a(f; n) = dc(A000290(n), A000010(n) * dc(A008683(n), f(n)) where dc(x, y) is Dirichlet convolution of x and y; if f is multiplicative, then a(f; n) is multiplicative; row sums of this triangle use f(n) = n (see formula section).

Examples

			The triangle T(n, k) for 1 <= k <= n starts:
n \k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   2   3
   3 :   3   5   5
   4 :   4   8   4   8
   5 :   5   9   9   9   9
   6 :   6  15  10   9  10  15
   7 :   7  13  13  13  13  13  13
   8 :   8  20   8  20   8  20   8  20
   9 :   9  21  21   9  21  21   9  21  21
  10 :  10  27  18  27  18  15  18  27  18  27
  11 :  11  21  21  21  21  21  21  21  21  21  21
  12 :  12  40  20  24  20  40  12  40  20  24  20  40
  etc.

Crossrefs

Cf. A000010, A000290, A008683, A018804, A127473.

Formula

T(n, 1) = n; T(n, n) = A018804(n).
T(n, k) = T(n, n+2-k) for 1 < k <= n.
Conjecture: Row sums equal Dirichlet convolution of A000290 and A127473.

A375745 a(n) is the sum of the vector of the reduced discriminant of the n-th cyclotomic polynomial.

Original entry on oeis.org

1, 1, 4, 4, 16, 4, 36, 16, 36, 16, 100, 16, 144, 36, 72, 64, 256, 36, 324, 64, 156, 100, 484, 64, 400, 144, 324, 144, 784, 72, 900, 256, 420, 256, 648, 144, 1296, 324, 600, 256, 1600, 156, 1764, 400, 648, 484, 2116, 256, 1764, 400, 1056, 576, 2704, 324, 1720, 576
Offset: 1

Views

Author

Darío Clavijo, Aug 26 2024

Keywords

Comments

Conjecture: a(n) >= phi(n)^2 = A127473(n), and for n>=2 strictly greater iff n is in A070537.

Crossrefs

Cf. A127473, A070537, A004124.

Programs

  • PARI
    a(n) = vecsum(poldiscreduced(polcyclo(n)));
Previous Showing 21-26 of 26 results.

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