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A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

%I A141809 #39 Feb 16 2025 08:33:08
%S A141809 1,2,3,4,5,2,3,7,8,9,2,5,11,4,3,13,2,7,3,5,16,17,2,9,19,4,5,3,7,2,11,
%T A141809 23,8,3,25,2,13,27,4,7,29,2,3,5,31,32,3,11,2,17,5,7,4,9,37,2,19,3,13,
%U A141809 8,5,41,2,3,7,43,4,11,9,5,2,23,47,16,3,49,2,25,3,17,4,13,53,2,27,5,11,8,7,3
%N A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).
%C A141809 In other words, except for row 1, row n contains the unitary prime power divisors of n, sorted by the prime. - _Franklin T. Adams-Watters_, May 05 2011
%C A141809 A034684(n) = smallest term of n-th row; A028233(n) = T(n,1); A053585(n) = T(n,A001221(n)); A008475(n) = sum of n-th row for n > 1. - _Reinhard Zumkeller_, Jan 29 2013
%H A141809 Reinhard Zumkeller, <a href="/A141809/b141809.txt">Rows n=1..10000 of triangle, flattened</a>
%H A141809 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactorization.html">Prime Factorization</a>
%F A141809 T(n,k) = A027748(n,k)^A124010(n,k) for n > 1, k = 1..A001221(n). - _Reinhard Zumkeller_, Mar 15 2012
%e A141809 60 has the prime factorization 2^2 * 3^1 * 5^1, so row 60 is (4,3,5).
%e A141809 From _M. F. Hasler_, Oct 12 2018: (Start)
%e A141809 The table starts:
%e A141809     n : largest prime powers dividing n
%e A141809     1 :  1
%e A141809     2 :  2
%e A141809     3 :  3
%e A141809     4 :  4
%e A141809     5 :  5
%e A141809     6 :  2, 3
%e A141809     7 :  7
%e A141809     8 :  8
%e A141809     9 :  9
%e A141809    10 :  2, 5
%e A141809    11 : 11
%e A141809    12 :  4, 3
%e A141809    etc. (End)
%t A141809 f[{x_, y_}] := x^y; Table[Map[f, FactorInteger[n]], {n, 1, 50}] // Grid (* _Geoffrey Critzer_, Apr 03 2015 *)
%o A141809 (Haskell)
%o A141809 a141809 n k = a141809_row n !! (k-1)
%o A141809 a141809_row 1 = [1]
%o A141809 a141809_row n = zipWith (^) (a027748_row n) (a124010_row n)
%o A141809 a141809_tabf = map a141809_row [1..]
%o A141809 -- _Reinhard Zumkeller_, Mar 18 2012
%o A141809 (PARI) A141809_row(n)=if(n>1, [f[1]^f[2]|f<-factor(n)~], [1]) \\ _M. F. Hasler_, Oct 12 2018, updated Aug 19 2022
%Y A141809 A027748, A124010 are used in a formula defining this sequence.
%Y A141809 Cf. A001221 (row lengths), A008475 (row sums), A028233 (column 1), A034684 (row minima), A053585 (right edge).
%Y A141809 Cf. A060175, A141810, A213925.
%K A141809 nonn,tabf
%O A141809 1,2
%A A141809 _Leroy Quet_, Jul 07 2008