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A027751 Irregular triangle read by rows in which row n lists the proper divisors of n (those divisors of n which are < n), with the first row {1} by convention.

%I A027751 #42 Jul 08 2025 17:55:04
%S A027751 1,1,1,1,2,1,1,2,3,1,1,2,4,1,3,1,2,5,1,1,2,3,4,6,1,1,2,7,1,3,5,1,2,4,
%T A027751 8,1,1,2,3,6,9,1,1,2,4,5,10,1,3,7,1,2,11,1,1,2,3,4,6,8,12,1,5,1,2,13,
%U A027751 1,3,9,1,2,4,7,14,1,1,2,3,5,6,10,15,1,1,2,4,8,16,1,3,11,1,2,17,1,5,7,1,2,3,4,6,9,12,18
%N A027751 Irregular triangle read by rows in which row n lists the proper divisors of n (those divisors of n which are < n), with the first row {1} by convention.
%C A027751 Or, take the list 1,2,3,4,... of natural numbers (A000027) and replace each number by its proper divisors.
%C A027751 The row length is 1 for n = 1 and A032741(n) for n >= 2. - _Wolfdieter Lang_, Jan 16 2016
%H A027751 Alois P. Heinz, <a href="/A027751/b027751.txt">Rows n = 1..1540, flattened</a>
%e A027751 The irregular triangle T(n,k) begins:
%e A027751 n\k  1 2 3 4  5 ...
%e A027751 1:   1  (by convention)
%e A027751 2:   1
%e A027751 3:   1
%e A027751 4:   1 2
%e A027751 5:   1
%e A027751 6:   1 2 3
%e A027751 7:   1
%e A027751 8:   1 2 4
%e A027751 9:   1 3
%e A027751 10:  1 2 5
%e A027751 11:  1
%e A027751 12:  1 2 3 4  6
%e A027751 13:  1
%e A027751 14:  1 2 7
%e A027751 15:  1 3 5
%e A027751 16:  1 2 4 8
%e A027751 17:  1
%e A027751 18:  1 2 3 6  9
%e A027751 19:  1
%e A027751 20:  1 2 4 5 10
%e A027751 .... reformatted - _Wolfdieter Lang_, Jan 16 2016
%p A027751 with(numtheory):
%p A027751 T:= n-> sort([(divisors(n) minus {n})[]])[]: T(1):=1:
%p A027751 seq(T(n), n=1..50); # _Alois P. Heinz_, Apr 11 2012
%t A027751 Table[ Divisors[n] // Most, {n, 1, 36}] // Flatten // Prepend[#, 1] & (* _Jean-François Alcover_, Jun 10 2013 *)
%o A027751 (Haskell)
%o A027751 a027751 n k = a027751_tabf !! (n-1) !! (k-1)
%o A027751 a027751_row n = a027751_tabf !! (n-1)
%o A027751 a027751_tabf = [1] : map init (tail a027750_tabf)
%o A027751 -- _Reinhard Zumkeller_, Apr 18 2012
%o A027751 (Python)
%o A027751 from sympy import divisors
%o A027751 def a(n): return [1] if n==1 else divisors(n)[:-1]
%o A027751 for n in range(21): print(a(n)) # _Indranil Ghosh_, Apr 30 2017
%o A027751 (PARI) row(n) = if (n==1, [1], my(d = divisors(n)); vector(#d-1,k, d[k])); \\ _Michel Marcus_, Apr 30 2017
%Y A027751 Cf. A027750, A032741 (row lengths), A001065, A000005.
%Y A027751 Row sums give A173455. - _Omar E. Pol_, Nov 23 2010
%K A027751 nonn,easy,tabf
%O A027751 1,5
%A A027751 _N. J. A. Sloane_
%E A027751 More terms from _Patrick De Geest_, May 15 1998
%E A027751 Example edited by _Omar E. Pol_, Nov 23 2010