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Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).
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%I #33 Oct 04 2024 08:51:45

%S 0,1,0,2,1,0,3,3,1,0,4,5,5,1,0,5,9,7,7,1,0,6,7,25,11,11,1,0,7,15,11,

%T 49,13,13,1,0,8,11,35,13,121,17,17,1,0,9,27,13,77,17,169,19,19,1,0,10,

%U 25,125,17,143,19,289,23,23,1,0,11,21,49,343,19,221,23,361,29,29,1,0

%N Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

%C Each row k is a multiplicative function, being in essence "the k-th power" of A003961, i.e., A(row,col) = A003961^row (col). Zeroth power gives an identity function, A001477, which occurs as the row zero.

%C The terms in the same column have the same prime signature.

%C The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

%H Antti Karttunen, <a href="/A242378/b242378.txt">Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened</a>

%F A(0,n) = n, A(row,0) = 0, A(row>0,n>0) = A003961(A(row-1,n)).

%e The top-left corner of the array:

%e 0, 1, 2, 3, 4, 5, 6, 7, 8, ...

%e 0, 1, 3, 5, 9, 7, 15, 11, 27, ...

%e 0, 1, 5, 7, 25, 11, 35, 13, 125, ...

%e 0, 1, 7, 11, 49, 13, 77, 17, 343, ...

%e 0, 1, 11, 13, 121, 17, 143, 19,1331, ...

%e 0, 1, 13, 17, 169, 19, 221, 23,2197, ...

%e ...

%e A(2,6) = A003961(A003961(6)) = p_{1+2} * p_{2+2} = p_3 * p_4 = 5 * 7 = 35, because 6 = 2*3 = p_1 * p_2.

%o (Scheme, with function factor from with Aubrey Jaffer's SLIB Scheme library)

%o (require 'factor)

%o (define (ifactor n) (cond ((< n 2) (list)) (else (sort (factor n) <))))

%o (define (A242378 n) (A242378bi (A002262 n) (A025581 n)))

%o (define (A242378bi row col) (if (zero? col) col (apply * (map A000040 (map (lambda (k) (+ k row)) (map A049084 (ifactor col)))))))

%Y Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.

%Y Transpose: A242379.

%Y Row 0: A001477, Row 1: A003961 (from 1 onward), Row 2: A357852 (from 1 onward), Row 3: A045968 (from 7 onward), Row 4: A045970 (from 11 onward).

%Y Column 2: A000040 (primes), Column 3: A065091 (odd primes), Column 4: A001248 (squares of primes), Column 6: A006094 (products of two successive primes), Column 8: A030078 (cubes of primes).

%Y Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

%K nonn,tabl

%O 0,4

%A _Antti Karttunen_, May 12 2014