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

%I A027750 #135 Jun 04 2025 00:34:56
%S A027750 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 A027750 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 A027750 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 A027750 Triangle read by rows in which row n lists the divisors of n.
%C A027750 Or, in the list of natural numbers (A000027), replace n with its divisors.
%C A027750 This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.
%C A027750 Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - _Omar E. Pol_, Sep 17 2008
%C A027750 Concatenation of n-th row gives A037278(n). - _Reinhard Zumkeller_, Aug 07 2011
%C A027750 {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 A027750 Row sums give A000203. Right border gives A000027. - _Omar E. Pol_, Jul 29 2012
%C A027750 Indices of records are in A006218. - _Irina Gerasimova_, Feb 27 2013
%C A027750 The number of primes in the n-th row is omega(n) = A001221(n). - _Michel Marcus_, Oct 21 2015
%C A027750 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
%C A027750 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
%C A027750 Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - _Ali Sada_, May 23 2025
%C A027750 All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - _Peter Luschny_, Jun 02 2025
%H A027750 Franklin T. Adams-Watters, <a href="/A027750/b027750.txt">Rows 1..1000, flattened</a>
%H A027750 Franklin T. Adams-Watters, <a href="/A027750/a027750.txt">Rows 1..10000</a>
%H A027750 Peter Luschny, <a href="/A027750/a027750_1.txt">A Python program using gcd for n >= 0</a>.
%H A027750 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv05.jpg">Illustration of initial terms</a>, (2009).
%H A027750 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Divisor.html">Divisor</a>
%H A027750 Wikipedia, <a href="http://www.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>
%H A027750 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%F A027750 a(A006218(n-1) + k) = k-divisor of n, 1 <= k <= A000005(n). - _Reinhard Zumkeller_, May 10 2006
%F A027750 T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - _Reinhard Zumkeller_, Sep 28 2014
%e A027750 Triangle begins:
%e A027750   1;
%e A027750   1, 2;
%e A027750   1, 3;
%e A027750   1, 2, 4;
%e A027750   1, 5;
%e A027750   1, 2, 3, 6;
%e A027750   1, 7;
%e A027750   1, 2, 4, 8;
%e A027750   1, 3, 9;
%e A027750   1, 2, 5, 10;
%e A027750   1, 11;
%e A027750   1, 2, 3, 4, 6, 12;
%e A027750   ...
%e A027750 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
%p A027750 seq(op(numtheory:-divisors(a)), a = 1 .. 20) # _Matt C. Anderson_, May 15 2017
%t A027750 Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]
%o A027750 (Magma) [Divisors(n) : n in [1..20]];
%o A027750 (Haskell)
%o A027750 a027750 n k = a027750_row n !! (k-1)
%o A027750 a027750_row n = filter ((== 0) . (mod n)) [1..n]
%o A027750 a027750_tabf = map a027750_row [1..]
%o A027750 -- _Reinhard Zumkeller_, Jan 15 2011, Oct 21 2010
%o A027750 (PARI) v=List();for(n=1,20,fordiv(n,d,listput(v,d)));Vec(v) \\ _Charles R Greathouse IV_, Apr 28 2011
%o A027750 (Python)
%o A027750 from sympy import divisors
%o A027750 for n in range(1, 16):
%o A027750     print(divisors(n)) # _Indranil Ghosh_, Mar 30 2017
%Y A027750 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.
%K A027750 nonn,easy,tabf,look
%O A027750 1,3
%A A027750 _N. J. A. Sloane_
%E A027750 More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)