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%I #63 Aug 14 2022 10:20:54
%S 0,1,0,1,1,0,2,2,1,0,1,2,3,1,0,2,4,2,4,1,0,1,3,9,2,5,1,0,3,8,4,16,2,6,
%T 1,0,2,3,27,5,25,2,7,1,0,2,4,3,64,6,36,2,8,1,0,1,5,6,3,125,7,49,2,9,1,
%U 0,3,16,10,8,3,216,8,64,2,10,1,0,1,4,81,17,10,3,343,9,81,2,11,1,0,2,32,5
%N Suppose m = Product_{i=1..k} p_i^e_i, where p_i is the i-th prime number and each e_i is a nonnegative integer. Then we can define P_m(x) = Sum_{i=1..k} e_i*x^(i-1). The sequence is the square array A(n,m) = P_m(n) read by descending antidiagonals.
%C From _Antti Karttunen_, Jul 29 2015: (Start)
%C The square array A(row,col) is read by downwards antidiagonals as: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
%C A(n,m) (entry at row=n, column=m) gives the evaluation at x=n of the polynomial (with nonnegative integer coefficients) bijectively encoded in the prime factorization of m. See A206284, A206296 for the details of that encoding. (The roles of variables n and m were accidentally swapped in this description, corrected by _Antti Karttunen_, Oct 30 2016)
%C (End)
%C Each row is a completely additive sequence, row n mapping prime(m) to n^(m-1). - _Peter Munn_, Apr 22 2022
%H Antti Karttunen, <a href="/A104244/b104244.txt">Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of the array</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A(n,A206296(k)) = A073133(n,k). [This formula demonstrates how this array can be used with appropriately encoded polynomials. Note that A073133 reads its antidiagonals by ascending order, while here the order is opposite.] - _Antti Karttunen_, Oct 30 2016
%F From _Peter Munn_, Apr 05 2021: (Start)
%F The sequence is defined by the following identities:
%F A(n, 3) = n;
%F A(n, m*k) = A(n, m) + A(n, k);
%F A(n, A297845(m, k)) = A(n, m) * A(n, k).
%F (End)
%e a(13) = 3 because 3 = p_1^0 * p_2^1 * p_3^0 * ..., so P_3(x) = 0*x^(1-1) + 1*x^(2-1) + 0*x^(3-1) + ... = x. Hence a(13) = A(3,3) = P_3(3) = 3. [Elaborated by _Peter Munn_, Aug 13 2022]
%e The top left corner of the array:
%e 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4
%e 0, 1, 2, 2, 4, 3, 8, 3, 4, 5, 16, 4, 32, 9, 6, 4
%e 0, 1, 3, 2, 9, 4, 27, 3, 6, 10, 81, 5, 243, 28, 12, 4
%e 0, 1, 4, 2, 16, 5, 64, 3, 8, 17, 256, 6, 1024, 65, 20, 4
%e 0, 1, 5, 2, 25, 6, 125, 3, 10, 26, 625, 7, 3125, 126, 30, 4
%e 0, 1, 6, 2, 36, 7, 216, 3, 12, 37, 1296, 8, 7776, 217, 42, 4
%e 0, 1, 7, 2, 49, 8, 343, 3, 14, 50, 2401, 9, 16807, 344, 56, 4
%e 0, 1, 8, 2, 64, 9, 512, 3, 16, 65, 4096, 10, 32768, 513, 72, 4
%e 0, 1, 9, 2, 81, 10, 729, 3, 18, 82, 6561, 11, 59049, 730, 90, 4
%e 0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4
%e ...
%o (MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
%o (require 'factor)
%o (define (A104244 n) (A104244bi (A002260 n) (A004736 n)))
%o (define (A104244bi row col) (fold-left (lambda (sum p.e) (+ sum (* (cdr p.e) (expt row (- (A000720 (car p.e)) 1))))) 0 (if (= 1 col) (list) (elemcountpairs (sort (factor col) <)))))
%o (define (elemcountpairs lista) (let loop ((pairs (list)) (lista lista) (prev #f)) (cond ((not (pair? lista)) (reverse! pairs)) ((equal? (car lista) prev) (set-cdr! (car pairs) (+ 1 (cdar pairs))) (loop pairs (cdr lista) prev)) (else (loop (cons (cons (car lista) 1) pairs) (cdr lista) (car lista))))))
%o ;; _Antti Karttunen_, Jul 29 2015
%Y Cf. A000720.
%Y Transpose: A104245.
%Y Main diagonal: A090883.
%Y Row 1: A001222, row 2: A048675, row 3: A090880, row 4: A090881, row 5: A090882, row 10: A054841; and, in the extrapolated table, row 0: A007814, row -1: A195017.
%Y Other completely additive sequences with prime(k) mapped to a function of k include k: A056239, k-1: A318995, k+1: A318994, k^2: A289506, 2^k-1: A293447, k!: A276075, F(k-1): A265753, F(k-2): A265752.
%Y For completely additive sequences with primes p mapped to a function of p, see A001414.
%Y For completely additive sequences where some primes are mapped to 1, the rest to 0 (notably, some ruler functions) see the cross-references in A249344.
%Y For completely additive sequences, s, with primes p mapped to a function of s(p-1) and maybe s(p+1), see A352957.
%Y See the formula section for the relationship to A073133, A206296.
%Y See the comments for the relevance of A206284.
%Y A297845 represents multiplication of the relevant polynomials.
%Y Cf. A090884, A248663, A265398, A265399 for other related sequences.
%Y A167219 lists columns that contain their own column number.
%K easy,nonn,tabl
%O 1,7
%A _Olaf Voß_, Feb 26 2005
%E Starting offset changed from 0 to 1 by _Antti Karttunen_, Jul 29 2015
%E Name edited (and aligned with rest of sequence) by _Peter Munn_, Apr 23 2022
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