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A127093
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Triangle read by rows: T(n,k)=k if k is a divisor of n; otherwise, T(n,k)=0 (1<=k<=n).
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38
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1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 2, 0, 4, 0, 0, 0, 8, 1, 0, 3, 0, 0, 0, 0, 0, 9, 1, 2, 0, 0, 5, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Sum of terms in row n = sigma(n) (sum of divisors of n, A000203(n)).
Euler's derivation of A127093 in polynomial form is in his proof of the formula for Sigma(n): (let S=Sigma, then Euler proved that S(n) = S(n-1) + S(n-2) - S(n-5) - S(n-7) + S(n-12) + S(n-15) - S(n-22) - S(n-26),...).
[Young, p. 365-366], Euler begins, s = (1-x)*(1-x^2)*(1-x^3)...= 1 - x - x^2 + x^5 + x^7 - x^12...; log s = log(1-x) + log(1-x^2) + log(1-x^3)...; differentiating and then changing signs, Euler has t = x/(1-x) + 2x^2/(1-x^2) + 3x^3/(1-x^3) + 4x^4/(1-x^4) + 5x^5/(1-x^5)...
Finally, Euler expands each term of t into a geometric series, getting A127093 in polynomial form: t =
x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + ...
..+2x^2..........+ 2x^4..........+ 2x^6.........+ 2x^8 + ...
.........+ 3x^3..................+ 3x^6.................+ ...
.................+ 4x^4.........................+ 4x^8..+ ...
.........................+ 5x^5.........................+ ...
.................................+ 6x^6.................+ ...
.........................................+ 7x^7.........+ ...
................................................+ 8x^8..+ ...
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REFERENCES
| David Wells, "Prime Numbers, the Most Mysterious Figures in Math", John Wiley & Sons, 2005, appendix.
L. Euler, "Discovery of a Most Extraordinary Law of the Numbers Concerning the Sum of Their Divisors"; pp. 358-367 of Robert M. Young, "Excursions in Calculus, An Interplay of the Continuous and the Discrete", MAA, 1992. See p. 366.
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LINKS
| Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
Eric Weisstein's World of Mathematics, Divisor
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FORMULA
| k-th column is composed of "k" interspersed with (k-1) zeros.
Let M = A127093 as an infinite lower triangular matrix and V = the harmonic series as a vector: [1/1, 1/2, 1/3,...]. then M*V = d(n), A000005: [1, 2, 2, 3, 2, 4, 2, 4, 3, 4,...]. M^2 * V = A060640: [1, 5, 7, 17, 11, 35, 15, 49, 34, 55,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 10 2007
Also mod(n-1;k) - mod(n;k) + 1 (1<=k<=n) - Mats Granvik (mgranvik(AT)abo.fi), Aug 31 2007
T(n,k) = k * 0 ^ (n mod k). [Reinhard Zumkeller, Jan 15 2011]
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EXAMPLE
| T(8,4) = 4 since 4 divides 8.
T(9,3) = 3 since 3 divides 9.
First few rows of the triangle are:
1;
1, 2;
1, 0, 3;
1, 2, 0, 4;
1, 0, 0, 0, 5;
1, 2, 3, 0, 0, 6;
1, 0, 0, 0, 0, 0, 7;
1, 2, 0, 4, 0, 0, 0, 8;
1, 0, 3, 0, 0, 0, 0, 0, 9;
...
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MAPLE
| A127093:=proc(n, k) if type(n/k, integer)=true then k else 0 fi end:
for n from 1 to 16 do seq(A127093(n, k), k=1..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 20 2007
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PROG
| (Excel cell formula) mod(row()-1; column()) - mod(row(); column()) + 1 - Mats Granvik (mgranvik(AT)abo.fi), Aug 31 2007
(Haskell)
a127093 n k = a127093_row n !! (k-1)
a127093_row n = zipWith (*) [1..n] $ map ((0 ^) . (mod n)) [1..n]
a127093_tabl = map a127093_row [1..]
-- Reinhard Zumkeller, Jan 15 2011
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CROSSREFS
| Reversal = A127094
Cf. A127094, A127095, A127096, A127097, A127098, A127099, A000203, A126988, A127013, A127057, A038040, A024916, A060640, A001001.
Cf. A000005, A060640.
Cf. A027750.
Sequence in context: A143256 A143151 A130106 * A141543 A182720 A146540
Adjacent sequences: A127090 A127091 A127092 * A127094 A127095 A127096
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 05 2007, Apr 04 2007
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 20 2007
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