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%I #53 Apr 30 2020 13:37:26
%S 1,2,3,2,5,2,3,7,2,3,2,5,11,2,3,13,2,7,3,5,2,17,2,3,19,2,5,3,7,2,11,
%T 23,2,3,5,2,13,3,2,7,29,2,3,5,31,2,3,11,2,17,5,7,2,3,37,2,19,3,13,2,5,
%U 41,2,3,7,43,2,11,3,5,2,23,47,2,3,7,2,5,3,17,2,13,53,2,3,5,11,2,7,3,19,2,29,59,2,3,5,61,2,31
%N Irregular triangle in which first row is 1, n-th row (n > 1) lists distinct prime factors of n.
%C Number of terms in n-th row is A001221(n) for n > 1.
%C From _Reinhard Zumkeller_, Aug 27 2011: (Start)
%C A008472(n) = Sum_{k=1..A001221(n)} T(n,k), n>1;
%C A007947(n) = Product_{k=1..A001221(n)} T(n,k);
%C A006530(n) = Max_{k=1..A001221(n)} T(n,k).
%C A020639(n) = Min_{k=1..A001221(n)} T(n,k).
%C (End)
%C Subsequence of A027750 that lists the divisors of n. - _Michel Marcus_, Oct 17 2015
%H T. D. Noe, <a href="/A027748/b027748.txt">Rows n=1..2048 of triangle, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>.
%e Triangle begins:
%e 1;
%e 2;
%e 3;
%e 2;
%e 5;
%e 2, 3;
%e 7;
%e 2;
%e 3;
%e 2, 5;
%e 11;
%e 2, 3;
%e 13;
%e 2, 7;
%e ...
%p with(numtheory): [ seq(factorset(n), n=1..100) ];
%t Flatten[ Table[ FactorInteger[n][[All, 1]], {n, 1, 62}]](* _Jean-François Alcover_, Oct 10 2011 *)
%o (Haskell)
%o import Data.List (unfoldr)
%o a027748 n k = a027748_tabl !! (n-1) !! (k-1)
%o a027748_tabl = map a027748_row [1..]
%o a027748_row 1 = [1]
%o a027748_row n = unfoldr fact n where
%o fact 1 = Nothing
%o fact x = Just (p, until ((> 0) . (`mod` p)) (`div` p) x)
%o where p = a020639 x -- smallest prime factor of x
%o -- _Reinhard Zumkeller_, Aug 27 2011
%o (PARI) print1(1);for(n=2,20,f=factor(n)[,1];for(i=1,#f,print1(", "f[i]))) \\ _Charles R Greathouse IV_, Mar 20 2013
%o (Python)
%o from sympy import primefactors
%o for n in range(2, 101):
%o print([i for i in primefactors(n)]) # _Indranil Ghosh_, Mar 31 2017
%Y Cf. A000027, A001221, A001222 (with repetition), A027746, A141809, A141810.
%Y a(A013939(A000040(n))+1) = A000040(n).
%Y Cf. A020639, A027750.
%Y A284411 gives column medians.
%K nonn,easy,tabf,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)
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