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 A027746 Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition. 98

%I

%S 1,2,3,2,2,5,2,3,7,2,2,2,3,3,2,5,11,2,2,3,13,2,7,3,5,2,2,2,2,17,2,3,3,

%T 19,2,2,5,3,7,2,11,23,2,2,2,3,5,5,2,13,3,3,3,2,2,7,29,2,3,5,31,2,2,2,

%U 2,2,3,11,2,17,5,7,2,2,3,3,37,2,19,3,13,2,2,2,5,41,2,3,7,43,2,2,11,3,3,5

%N Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition.

%C n-th row has length A001222(n) (n>1).

%C A001414(n)=Sum(T(n,k):1<=k<=A001222(n)), n>1; A006530(n) = T(n,A001222(n)) = Max(T(n,k):1<=k<=A001222(n)); A020639(n) = T(n,1) = Min(T(n,k):1<=k<=A001222(n)). [_Reinhard Zumkeller_, Aug 27 2011]

%H N. J. A. Sloane, <a href="/A027746/b027746.txt">First 2048 rows of triangle, flattened</a>

%H S. von Worley (?), <a href="http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/">Animated Factorization Diagrams</a>, Oct. 2012.

%F Product(T(n,k): 1 <= k <= A001221(n)) = n.

%e Triangle begins

%e 1;

%e 2;

%e 3;

%e 2, 2;

%e 5;

%e 2, 3;

%e 7;

%e 2, 2, 2;

%e 3, 3;

%e 2, 5;

%e ...

%p P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1],k=1..FM[j][2]),j=1..nops(FM)) end: 1; for n from 2 to 45 do P(n) od; # yields sequence in triangular form; _Emeric Deutsch_, Feb 13 2005

%t row[n_] := Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; Flatten[ Table[ row[n], {n, 1, 45}]] (* _Jean-François Alcover_, Dec 01 2011 *)

%o import Data.List (unfoldr)

%o a027746 n k = a027746_tabl !! (n-1) !! (k-1)

%o a027746_tabl = map a027746_row [1..]

%o a027746_row 1 = [1]

%o a027746_row n = unfoldr fact n where

%o fact 1 = Nothing

%o fact x = Just (p, x `div` p) where p = a020639 x

%o -- _Reinhard Zumkeller_, Aug 27 2011

%o (PARI) A027746_row(n,o=[1])=if(n>1,concat(apply(t->vector(t[2],i,t[1]), Vec(factor(n)~))),o) \\ Use %(n,[]) if you want the more natural [] for the first row. - _M. F. Hasler_, Jul 29 2015

%o (Sage) v=[1]

%o for k in [2..45]: v=add(([p] * m for (p, m) in factor(k)),v)

%o print v # _Giuseppe Coppoletta_, Dec 29 2017

%Y Cf. A000027, A001222, A027748.

%Y a(A022559(A000040(n))+1) = A000040(n).

%Y Column 1 is A020639, columns 2 and 3 correspond to A014673 and A115561.

%Y A281890 measures frequency of each prime in each column, with A281889 giving median values.

%Y Cf. A175943 (partial products), A265110 (partial row products), A265111.

%K nonn,easy,nice,tabf

%O 1,2