A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 5, 1, 0, 5, 9, 7, 7, 1, 0, 6, 7, 25, 11, 11, 1, 0, 7, 15, 11, 49, 13, 13, 1, 0, 8, 11, 35, 13, 121, 17, 17, 1, 0, 9, 27, 13, 77, 17, 169, 19, 19, 1, 0, 10, 25, 125, 17, 143, 19, 289, 23, 23, 1, 0, 11, 21, 49, 343, 19, 221, 23, 361, 29, 29, 1, 0

Offset

0,4

Comments

Each row k is a multiplicative function, being in essence "the k-th power" of A003961, i.e., A(row,col) = A003961^row (col). Zeroth power gives an identity function, A001477, which occurs as the row zero.

The terms in the same column have the same prime signature.

The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

Links

Antti Karttunen, Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened

Formula

A(0,n) = n, A(row,0) = 0, A(row>0,n>0) = A003961(A(row-1,n)).

Example

The top-left corner of the array:
  0,   1,   2,   3,   4,   5,   6,   7,   8, ...
  0,   1,   3,   5,   9,   7,  15,  11,  27, ...
  0,   1,   5,   7,  25,  11,  35,  13, 125, ...
  0,   1,   7,  11,  49,  13,  77,  17, 343, ...
  0,   1,  11,  13, 121,  17, 143,  19,1331, ...
  0,   1,  13,  17, 169,  19, 221,  23,2197, ...
...
A(2,6) = A003961(A003961(6)) = p_{1+2} * p_{2+2} = p_3 * p_4 = 5 * 7 = 35, because 6 = 2*3 = p_1 * p_2.

Prog

(Scheme, with function factor from with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (ifactor n) (cond ((< n 2) (list)) (else (sort (factor n) <))))
(define (A242378 n) (A242378bi (A002262 n) (A025581 n)))
(define (A242378bi row col) (if (zero? col) col (apply * (map A000040 (map (lambda (k) (+ k row)) (map A049084 (ifactor col)))))))

Crossrefs

Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.

Transpose: A242379.

Row 0: A001477, Row 1: A003961 (from 1 onward), Row 2: A357852 (from 1 onward), Row 3: A045968 (from 7 onward), Row 4: A045970 (from 11 onward).

Column 2: A000040 (primes), Column 3: A065091 (odd primes), Column 4: A001248 (squares of primes), Column 6: A006094 (products of two successive primes), Column 8: A030078 (cubes of primes).

Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

Sequence in context: A100224 A208671 A208727 * A268820 A199011 A206735

Adjacent sequences: A242375 A242376 A242377 * A242379 A242380 A242381

Keyword

nonn,tabl

Author

Antti Karttunen, May 12 2014

Status

approved