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A104244 Suppose n=(p1^e1)(p2^e2)... where p1,p2,... are the prime numbers and e1,e2,... are nonnegative integers. Then we can define Pn(x) = e1 + (e2)*x + (e3)*(x^2) + (e4)*(x^3) + ... + (ek)*(x^(k-1)) + ... The sequence is the table T(n,x)=Pn(x) read by descending antidiagonals. 8
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

From Antti Karttunen, Jul 29 2015: (Start)

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.

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)

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of the array

Index entries for sequences computed from exponents in factorization of n

FORMULA

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

EXAMPLE

a(13)=3 because: 3=(p1^0)(p2^1)(p3^0)..., so P3(x)=x. Hence a(13) = T(3,3) = P3(3) = 3.

The top left corner of the array:

0, 1,  1, 2,   1,  2,   1,  3,  2,   2,     1,  3,      1,    2,   2, 4

0, 1,  2, 2,   4,  3,   8,  3,  4,   5,    16,  4,     32,    9,   6, 4

0, 1,  3, 2,   9,  4,  27,  3,  6,  10,    81,  5,    243,   28,  12, 4

0, 1,  4, 2,  16,  5,  64,  3,  8,  17,   256,  6,   1024,   65,  20, 4

0, 1,  5, 2,  25,  6,  125, 3, 10,  26,   625,  7,   3125,  126,  30, 4

0, 1,  6, 2,  36,  7,  216, 3, 12,  37,  1296,  8,   7776,  217,  42, 4

0, 1,  7, 2,  49,  8,  343, 3, 14,  50,  2401,  9,  16807,  344,  56, 4

0, 1,  8, 2,  64,  9,  512, 3, 16,  65,  4096, 10,  32768,  513,  72, 4

0, 1,  9, 2,  81, 10,  729, 3, 18,  82,  6561, 11,  59049,  730,  90, 4

0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4

...

PROG

(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)

(require 'factor)

(define (A104244 n) (A104244bi (A002260 n) (A004736 n)))

(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) <)))))

(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))))))

;; Antti Karttunen, Jul 29 2015

CROSSREFS

Cf. A000720.

Transpose: A104245.

Main diagonal: A090883.

Row 1: A001222, Row 2: A048675, Row 3: A090880, Row 4: A090881, Row 5: A090882, Row 10: A054841.

Cf. A090883, A090884, A206284, A206442, A248663, A265398, A265399, A265752, A265753, A276075, A276085, A277322 for other related sequences.

Cf. also A073133, A206296.

Sequence in context: A307011 A285007 A194527 * A116403 A123149 A185158

Adjacent sequences:  A104241 A104242 A104243 * A104245 A104246 A104247

KEYWORD

easy,nonn,tabl

AUTHOR

Olaf Voß, Feb 26 2005

EXTENSIONS

Starting offset changed from 0 to 1 by Antti Karttunen, Jul 29 2015

STATUS

approved

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Last modified October 15 13:06 EDT 2019. Contains 328030 sequences. (Running on oeis4.)