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A027750 Triangle read by rows in which row n lists the divisors of n. 328

%I

%S 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,

%T 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,

%U 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

%N Triangle read by rows in which row n lists the divisors of n.

%C Or, in the list of natural numbers (A000027), replace n with its divisors.

%C This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.

%C Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - _Omar E. Pol_, Sep 17 2008

%C Concatenation of n-th row gives A037278(n). - _Reinhard Zumkeller_, Aug 07 2011

%C {A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - _Reinhard Zumkeller_, Mar 18 2012

%C Row sums give A000203. Right border gives A000027. - _Omar E. Pol_, Jul 29 2012

%C Indices of records are in A006218. - _Irina Gerasimova_, Feb 27 2013

%C The number of primes in the n-th row is omega(n) = A001221(n). - _Michel Marcus_, Oct 21 2015

%C 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 in A292226. - _Wolfdieter Lang_, Nov 09 2017

%H Franklin T. Adams-Watters, <a href="/A027750/b027750.txt">Rows 1..1000, flattened</a>

%H Franklin T. Adams-Watters, <a href="/A027750/a027750.txt">Rows 1..10000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv05.jpg">Illustration of initial terms</a>, (2009).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Divisor.html">Divisor</a>

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>

%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>

%F a(A006218(n-1) + k) = k-divisor of n, 1 <= k <= A000005(n). - _Reinhard Zumkeller_, May 10 2006

%F T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - _Reinhard Zumkeller_, Sep 28 2014

%e Triangle begins:

%e 1;

%e 1, 2;

%e 1, 3;

%e 1, 2, 4;

%e 1, 5;

%e 1, 2, 3, 6;

%e 1, 7;

%e 1, 2, 4, 8;

%e 1, 3, 9;

%e 1, 2, 5, 10;

%e 1, 11;

%e 1, 2, 3, 4, 6, 12;

%p seq(op(numtheory:-divisors(a)), a = 1 .. 20) # _Matt C. Anderson_, May 15 2017

%t Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]

%o (MAGMA) [Divisors(n) : n in [1..20]];

%o (Haskell)

%o a027750 n k = a027750_row n !! (k-1)

%o a027750_row n = filter ((== 0) . (mod n)) [1..n]

%o a027750_tabf = map a027750_row [1..]

%o -- _Reinhard Zumkeller_, Jan 15 2011, Oct 21 2010

%o (PARI) v=List();for(n=1,20,fordiv(n,d,listput(v,d)));Vec(v) \\ _Charles R Greathouse IV_, Apr 28 2011

%o (Python)

%o from sympy import divisors

%o for n in xrange(1, 16):

%o ....print [i for i in divisors(n)] # _Indranil Ghosh_, Mar 30 2017

%Y 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.

%K nonn,easy,tabf,look

%O 1,3

%A _N. J. A. Sloane_

%E More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

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Last modified October 18 01:01 EDT 2018. Contains 316297 sequences. (Running on oeis4.)