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A027750
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Triangle read by rows in which row n lists the divisors of n.
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497
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1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 29
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OFFSET
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1,3
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COMMENTS
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Or, in the list of natural numbers (A000027), replace n with its divisors.
This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.
Also, row n lists the largest parts of the partitions of n whose parts are not distinct. -Omar E. Pol,Sep 17 2008
The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given inA292226.-Wolfdieter Lang,Nov 09 2017
T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). -Omar E. Pol,Nov 20 2019
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LINKS
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Eric Weisstein's World of Mathematics,Divisor
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FORMULA
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EXAMPLE
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Triangle begins:
1;
1, 2;
1, 3;
1, 2, 4;
1, 5;
1, 2, 3, 6;
1, 7;
1, 2, 4, 8;
1, 3, 9;
1, 2, 5, 10;
1, 11;
1, 2, 3, 4, 6, 12;
...
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of parts are [1, 2, 3, 6] respectively, the same as the divisors of 6. -Omar E. Pol,Nov 20 2019
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MAPLE
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MATHEMATICA
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Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]
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PROG
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(Magma) [Divisors(n): n in [1..20]];
(Haskell)
a027750 n k = a027750_row n!! (k-1)
a027750_row n = filter ((== 0). (mod n)) [1..n]
a027750_tabf = map a027750_row [1..]
(Python)
from sympy import divisors
for n in range(1, 16):
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CROSSREFS
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Cf.A000005(row length),A001221,A027749,A027751,A056534,A056538,A127093,A135010,A161700,A163280,A240698(partial sums of rows),A240694(partial products of rows),A247795(parities),A292226,A244051.
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)
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STATUS
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approved
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