A273673 Square array A(n,k) = (n / prime(1+A084558(k))^e) * prime(1+A084558(k)-A099563(k))^e, where e = A249344((1+A084558(k)), n) = the exponent of the largest power of prime(1+A084558(k)) which divides n. Array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

1, 1, 2, 1, 2, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 3, 4, 1, 2, 3, 4, 3, 6, 7, 1, 2, 3, 4, 2, 6, 7, 8, 1, 2, 3, 4, 2, 6, 7, 8, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 6, 11, 1, 2, 3, 4, 5, 6, 5, 8, 9, 6, 11, 8, 1, 2, 3, 4, 5, 6, 5, 8, 9, 4, 11, 12, 13, 1, 2, 3, 4, 5, 6, 5, 8, 9, 4, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 5, 8, 9, 10, 11, 12, 13, 14, 10

Offset: 1

Author

Antti Karttunen, Aug 09 2016

Keywords

• nonn
• base
• tabl

Comments

Informally: "clear" the exponent of prime(1+A084558(k)) and add it (the old value of exponent) to the exponent of prime(1+A084558(k)-A099563(k)) in the prime factorization of n.

Auxiliary function for computing array A275723.

Examples

			The top left 6 x 15 corner of the array:
   1,  1,  1,  1,  1,  1
   2,  2,  2,  2,  2,  2
   2,  3,  3,  3,  3,  3
   4,  4,  4,  4,  4,  4
   5,  3,  3,  2,  2,  5
   4,  6,  6,  6,  6,  6
   7,  7,  7,  7,  7,  5
   8,  8,  8,  8,  8,  8
   4,  9,  9,  9,  9,  9
  10,  6,  6,  4,  4, 10
  11, 11, 11, 11, 11, 11
   8, 12, 12, 12, 12, 12
  13, 13, 13, 13, 13, 13
  14, 14, 14, 14, 14, 10
  10,  9,  9,  6,  6, 15
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of array
• Indranil Ghosh, Python program for computing this sequence
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A000040, A084558, A099563, A249344.

Cf. also A007623, A275723.

Programs

Scheme

(define (A273673 n) (A273673bi (A002260 n) (A004736 n)))
(define (A273673bi n c) (if (zero? c) n (* (/ n (expt (A000040 (+ 1 (A084558 c))) (A249344bi (+ 1 (A084558 c)) n))) (expt (A000040 (+ 1 (- (A084558 c) (A099563 c)))) (A249344bi (+ 1 (A084558 c)) n)))))
;; Code for A249344bi given in A249344.

Formula

A(n,k) = (n / prime(1+A084558(k))^e) * prime(1+A084558(k)-A099563(k))^e, where e = A249344((1+A084558(k)), n), the exponent of the largest power prime(1+A084558(k)) which divides n.