cp's OEIS Frontend

A104244 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.

%I A104244 #63 Aug 14 2022 10:20:54
%S A104244 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 A104244 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 A104244 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 A104244 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 A104244 From _Antti Karttunen_, Jul 29 2015: (Start)
%C A104244 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 A104244 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 A104244 (End)
%C A104244 Each row is a completely additive sequence, row n mapping prime(m) to n^(m-1). - _Peter Munn_, Apr 22 2022
%H A104244 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 A104244 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A104244 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 A104244 From _Peter Munn_, Apr 05 2021: (Start)
%F A104244 The sequence is defined by the following identities:
%F A104244 A(n, 3) = n;
%F A104244 A(n, m*k) = A(n, m) + A(n, k);
%F A104244 A(n, A297845(m, k)) = A(n, m) * A(n, k).
%F A104244 (End)
%e A104244 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 A104244 The top left corner of the array:
%e A104244 0, 1,  1, 2,   1,  2,   1,  3,  2,   2,     1,  3,      1,    2,   2, 4
%e A104244 0, 1,  2, 2,   4,  3,   8,  3,  4,   5,    16,  4,     32,    9,   6, 4
%e A104244 0, 1,  3, 2,   9,  4,  27,  3,  6,  10,    81,  5,    243,   28,  12, 4
%e A104244 0, 1,  4, 2,  16,  5,  64,  3,  8,  17,   256,  6,   1024,   65,  20, 4
%e A104244 0, 1,  5, 2,  25,  6,  125, 3, 10,  26,   625,  7,   3125,  126,  30, 4
%e A104244 0, 1,  6, 2,  36,  7,  216, 3, 12,  37,  1296,  8,   7776,  217,  42, 4
%e A104244 0, 1,  7, 2,  49,  8,  343, 3, 14,  50,  2401,  9,  16807,  344,  56, 4
%e A104244 0, 1,  8, 2,  64,  9,  512, 3, 16,  65,  4096, 10,  32768,  513,  72, 4
%e A104244 0, 1,  9, 2,  81, 10,  729, 3, 18,  82,  6561, 11,  59049,  730,  90, 4
%e A104244 0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4
%e A104244 ...
%o A104244 (MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
%o A104244 (require 'factor)
%o A104244 (define (A104244 n) (A104244bi (A002260 n) (A004736 n)))
%o A104244 (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 A104244 (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 A104244 ;; _Antti Karttunen_, Jul 29 2015
%Y A104244 Cf. A000720.
%Y A104244 Transpose: A104245.
%Y A104244 Main diagonal: A090883.
%Y A104244 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 A104244 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 A104244 For completely additive sequences with primes p mapped to a function of p, see A001414.
%Y A104244 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 A104244 For completely additive sequences, s, with primes p mapped to a function of s(p-1) and maybe s(p+1), see A352957.
%Y A104244 See the formula section for the relationship to A073133, A206296.
%Y A104244 See the comments for the relevance of A206284.
%Y A104244 A297845 represents multiplication of the relevant polynomials.
%Y A104244 Cf. A090884, A248663, A265398, A265399 for other related sequences.
%Y A104244 A167219 lists columns that contain their own column number.
%K A104244 easy,nonn,tabl
%O A104244 1,7
%A A104244 _Olaf Voß_, Feb 26 2005
%E A104244 Starting offset changed from 0 to 1 by _Antti Karttunen_, Jul 29 2015
%E A104244 Name edited (and aligned with rest of sequence) by _Peter Munn_, Apr 23 2022