|
Search: id:A054523
|
|
|
| A054523 |
|
Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1<=k<=n). |
|
+0
31
|
|
| 1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 4, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 4, 4, 0, 0, 1, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 6
(list; table; graph; listen)
|
|
|
OFFSET
|
1,4
|
|
|
COMMENT
|
Comments from Gary Adamson, Jan 08 2007: (Start) Let H be this lower triangular matrix. Then:
H * A051731 = A126988,
H * [1, 2, 3,...] = 1, 3, 5, 8, 9, 15,...= A018804,
H * sigma(n) = A038040 = d(n) * n = 1, 4, 6, 12, 10,... where sigma(n) = A000203,
H * d(n) (A000005) = sigma(n) = A000203,
Row sums of H = A018804 = sum of GCD (k,n),
H^2 * d(n) = d(n)*n, H^2 = A127192,
H * mu(n) (A008683) = phi(n) = A000010,
H^2 row sums = A018804. (End)
The Mobius inversion principle of Richard Dedekind and Joseph Liouville (1857). Cf. "Concrete Mathematics", p. 136; is equivalent to the statement that row sums of triangle A054523 = n, where the triangle = A054525 * A126988. A054525 = the Mobius transform and A126988 records the divisors of n by rows. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 03 2008]
Row sums are: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,...}. [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 18 2008]
|
|
REFERENCES
|
"Concrete Mathematics", Ronald L. Graham, Donald E. Knuth, & Oren Patashnik; Addison-Wesley, 2-nd ed., 1994, p. 136. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 03 2008]
|
|
FORMULA
|
Equals A054525 * A126988 as infinite lower triangular matrices. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 03 2008]
Can be obtained from the necklace polynomials in Mathematica: p(x,n)=(n/x)*NecklacePolynomial[n, x, Cyclic]]; t(n,m)=coefficients(p(x,n)). [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 18 2008]
|
|
EXAMPLE
|
Triangle begins {1}, {1, 1}, {2, 0, 1}, {2, 1, 0, 1}, {4, 0, 0, 0, 1}, {2, 2, 1, 0, 0, 1}, {6, 0, 0, 0, 0, 0, 1}, {4, 2, 0, 1, 0, 0, 0, 1}, {6, 0, 2, 0, 0, 0, 0, 0, 1}, {4, 4, 0, 0, 1, 0, 0, 0, 0, 1}, {10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1} [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 18 2008]
|
|
MATHEMATICA
|
<< DiscreteMath`Combinatorica`; Table[ExpandAll[(n/x)*NecklacePolynomial[n, x, Cyclic]], {n, 1, 12}]; Table[CoefficientList[ExpandAll[(n/x)*NecklacePolynomial[n, x, Cyclic]], x], {n, 1, 12}]; Flatten[%] [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 18 2008]
|
|
CROSSREFS
|
Cf. A054521, A054525.
Adjacent sequences: A054520 A054521 A054522 this_sequence A054524 A054525 A054526
Sequence in context: A117170 A117466 A136266 this_sequence A161363 A106351 A096800
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2000
|
|
|
Search completed in 0.003 seconds
|
