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%I #25 Dec 03 2022 05:43:00
%S 1,1,1,1,2,1,1,4,3,1,1,8,9,6,1,1,16,27,36,5,1,1,32,81,216,25,10,1,1,
%T 64,243,1296,125,100,15,1,1,128,729,7776,625,1000,225,30,1,1,256,2187,
%U 46656,3125,10000,3375,900,7,1,1,512,6561,279936,15625,100000,50625,27000,49,14
%N Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.
%C The A019565 row order gives the table neat relationships with A003961, A003987, A059897, A225546, A319075 and A329050. See the formula section.
%C Transposition of this table, that is reflection about its main diagonal, has subtle symmetries. For example, consider the unique factorization of a number into powers of distinct primes. This can be restated as factorization into numbers from rows 2^n (n >= 0) with no more than one from each row. Reflecting about the main diagonal, this factorization becomes factorization (of a related number) into numbers from columns 2^k (k >= 0) with no more than one from each column. This is also unique and is factorization into powers of squarefree numbers with distinct exponents that are powers of two. See the example section.
%F A(n,k) = A019565(n)^k.
%F A(k,n) = A225546(A(n,k)).
%F A(n,2k) = A000290(A(n,k)) = A(n,k)^2.
%F A(2n,k) = A003961(A(n,k)).
%F A(n,2k+1) = A(n,2k) * A(n,1).
%F A(2n+1,k) = A(2n,k) * A(1,k).
%F A(A003987(n,m), k) = A059897(A(n,k), A(m,k)).
%F A(n, A003987(m,k)) = A059897(A(n,m), A(n,k)).
%F A(2^n,k) = A319075(k,n+1).
%F A(2^n, 2^k) = A329050(n,k).
%F A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)), = A329329(A(n,1), A(1,k)).
%F Sum_{n>=0} 1/A(n,k) = zeta(k)/zeta(2*k), for k >= 2. - _Amiram Eldar_, Dec 03 2022
%e Square array A(n,k) begins:
%e n\k | 0 1 2 3 4 5 6 7
%e ----+------------------------------------------------------------------
%e 0| 1 1 1 1 1 1 1 1
%e 1| 1 2 4 8 16 32 64 128
%e 2| 1 3 9 27 81 243 729 2187
%e 3| 1 6 36 216 1296 7776 46656 279936
%e 4| 1 5 25 125 625 3125 15625 78125
%e 5| 1 10 100 1000 10000 100000 1000000 10000000
%e 6| 1 15 225 3375 50625 759375 11390625 170859375
%e 7| 1 30 900 27000 810000 24300000 729000000 21870000000
%e 8| 1 7 49 343 2401 16807 117649 823543
%e 9| 1 14 196 2744 38416 537824 7529536 105413504
%e 10| 1 21 441 9261 194481 4084101 85766121 1801088541
%e 11| 1 42 1764 74088 3111696 130691232 5489031744 230539333248
%e 12| 1 35 1225 42875 1500625 52521875 1838265625 64339296875
%e Reflection of factorization about the main diagonal: (Start)
%e The canonical (prime power) factorization of 864 is 2^5 * 3^3 = 32 * 27. Reflecting the factors about the main diagonal of the table gives us 10 * 36 = 10^1 * 6^2 = 360. This is the unique factorization of 360 into powers of squarefree numbers with distinct exponents that are powers of two.
%e Reflection about the main diagonal is given by the self-inverse function A225546(.). Clearly, all positive integers are in the domain of A225546, whether or not they appear in the table. It is valid to start from 360, observe that A225546(360) = 864, then use 864 to derive 360's factorization into appropriate powers of squarefree numbers as above.
%e (End)
%Y The range of values is A072774.
%Y Rows (abbreviated list): A000079(1), A000244(2), A000400(3), A000351(4), A011557(5), A001024(6), A009974(7), A000420(8), A001023(9), A009965(10), A001020(16), A001022(32), A001026(64).
%Y A019565 is column 1, A334110 is column 2, and columns that are sorted in increasing order (some without the 1) are: A005117(1), A062503(2), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7).
%Y Other subtables: A182944, A319075, A329050.
%Y Re-ordered subtable of A297845, A306697, A329329.
%Y A000290, A003961, A003987, A059897 and A225546 are used to express relationships between terms of this sequence.
%Y Cf. A285322.
%K nonn,tabl
%O 0,5
%A _Peter Munn_, Nov 10 2019
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