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A054523
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Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1<=k<=n).
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31
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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; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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 are the row index n. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 03 2008]
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REFERENCES
| Ronald L. Graham, D. E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley, 2-nd ed., 1994, p. 136. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 03 2008]
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FORMULA
| Equals A054525 * A126988 as infinite lower triangular matrices. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 03 2008]
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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;
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MAPLE
| A054523 := proc(n, k) if n mod k = 0 then numtheory[phi](n/k) ; else 0; end if; end proc: # R. J. Mathar, Apr 11 2011
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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]
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CROSSREFS
| Cf. A054521, A054525.
Sequence in context: A117170 A117466 A136266 * A161363 A106351 A096800
Adjacent sequences: A054520 A054521 A054522 * A054524 A054525 A054526
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Apr 09 2000
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