A273673 -id:A273673 - cp's OEIS frontend

A275723 Square array A(n,k) = prime factorization of n (= 1..) completely reduced by factorial base representation of k (= 0..), read by descending antidiagonals as A(1,0), A(1,1), A(2,0), A(1,2), A(2,1), A(3,0), etc. See the Comments section for the meaning of reduction in this context.

1, 1, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 3, 4, 5, 1, 2, 2, 4, 5, 6, 1, 2, 3, 4, 3, 4, 7, 1, 2, 2, 4, 2, 6, 7, 8, 1, 2, 3, 4, 2, 4, 7, 8, 9, 1, 2, 2, 4, 2, 6, 7, 8, 4, 10, 1, 2, 3, 4, 5, 4, 7, 8, 9, 10, 11, 1, 2, 2, 4, 5, 6, 7, 8, 4, 6, 11, 12, 1, 2, 3, 4, 3, 4, 5, 8, 9, 4, 11, 8, 13, 1, 2, 2, 4, 2, 6, 5, 8, 4, 4, 11, 12, 13, 14, 1, 2, 3, 4, 2, 4, 3, 8, 9, 4, 11, 8, 13, 14, 15

Offset: 1

Antti Karttunen, Aug 09 2016

The reduction is done by scanning the factorial base representation of k [see A007623] from its most significant end [where the most significant digit occurs at the one-based position A084558(k)], by adding the exponent of prime(1+A084558(k)) of n to the current exponent (possibly zero) of prime(1+A084558(k)-A099563(k)) in the prime factorization of n, after which the exponent of prime(1+A084558(k)) is changed to zero. Thus the total number of prime factors of n [A001222(n)] never changes. This single step of reduction is performed with a bivariate function A273673. The reduction then proceeds to the next digit to the right, effectively skipping any zeros until all factorial base digits of k have been scanned through and the prime factorization of n has been changed accordingly. See the examples.

This bivariate function is used to compute A275725.

			The top left 7 X 15 corner of the array:
   1,  1,  1,  1,  1,  1,  1
   2,  2,  2,  2,  2,  2,  2
   3,  2,  3,  2,  3,  2,  3
   4,  4,  4,  4,  4,  4,  4
   5,  5,  3,  2,  2,  2,  5
   6,  4,  6,  4,  6,  4,  6
   7,  7,  7,  7,  7,  7,  5
   8,  8,  8,  8,  8,  8,  8
   9,  4,  9,  4,  9,  4,  9
  10, 10,  6,  4,  4,  4, 10
  11, 11, 11, 11, 11, 11, 11
  12,  8, 12,  8, 12,  8, 12
  13, 13, 13, 13, 13, 13, 13
  14, 14, 14, 14, 14, 14, 10
  15, 10,  9,  4,  6,  4, 15
For row 15 (above), we have 15 = 3*5 = prime(2)*prime(3) and the terms for columns 0 - 6 (in factorial base: 0, 1, 10, 11, 20, 21, 100, see A007623) are computed as:
When k=0, we do nothing and n stays as n (thus column 0 of array is A000027).
When k=1 (with the length 1), we transfer the exponent of prime(2) to prime(1), to get prime(1)*prime(3) = 2*5 = 10.
When k=2, in factorial base "10", with the length 2, we transfer (add) the exponent of prime(3) to prime(2), to get prime(2)*prime(2) = 9.
When k=3, in factorial base "11", we first do as above, to get 9 = prime(2)^2, and for the least significant one, we transfer (add) the exponent of prime(2) to prime(1), to get prime(1)*prime(1) = 4.
When k=4, in factorial base "20", with the length 2, we transfer (add) the exponent of prime(3) to prime(1), to get prime(2)*prime(1) = 6.
When k=5, in factorial base "21", we first do as above, to get 6 = prime(2)*prime(1), and for the remaining "1" in factorial base representation of k, we transfer (add) the exponent of prime(2) to prime(1), to get prime(1)*prime(1) = 4.
When k=6, in factorial base "100", with the length 3, we transfer (add) the exponent of prime(4) to prime(3), but prime(4) = 7 whose exponent is zero in 15, thus the result is also 15.

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

Transpose: A275724.
Column 0: A000027.
Cf. A001222, A002260, A004736, A007623, A257687, A273673, A275725.

(define (A275723 n) (A275723bi (A002260 n) (- (A004736 n) 1)))
(define (A275723bi n fex) (let loop ((n n) (fex fex)) (cond ((zero? fex) n) (else (loop (A273673bi n fex) (A257687 fex))))))
;; Code for A273673bi given in A273673.

A(n,0) = n, and for k > 0, A(n,k) = A(A273673(n,k), A257687(k)).
Other identities. For all n >= 1 and k >= 0:
A001222(A(n,k)) = A001222(n). [This reduction doesn't change the total number of prime factors of n.]

cp's OEIS Frontend

A275723 Square array A(n,k) = prime factorization of n (= 1..) completely reduced by factorial base representation of k (= 0..), read by descending antidiagonals as A(1,0), A(1,1), A(2,0), A(1,2), A(2,1), A(3,0), etc. See the Comments section for the meaning of reduction in this context.

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