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

1,3

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.

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

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.

Example

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

Prog

(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.

Crossrefs

Cf. A000040, A084558, A099563, A249344.

Cf. also A007623, A275723.

Sequence in context: A131400 A132749 A271106 * A376570 A140186 A078498

Adjacent sequences: A273670 A273671 A273672 * A273674 A273675 A273676

Keyword

nonn,base,tabl

Author

Antti Karttunen, Aug 09 2016

Status

approved