This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A242378 #39 Sep 02 2025 09:24:04
%S A242378 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,
%T A242378 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,
%U A242378 25,125,17,143,19,289,23,23,1,0,11,21,49,343,19,221,23,361,29,29,1,0
%N 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).
%C A242378 Each row i is a multiplicative function, being in essence "the i-th power" of A003961, i.e., A(i,j) = A003961^i (j). Zeroth power gives an identity function, A001477, which occurs as the row zero.
%C A242378 The terms in the same column have the same prime signature.
%C A242378 The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .
%H A242378 Antti Karttunen, <a href="/A242378/b242378.txt">Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened</a>
%F A242378 A(0,j) = j, A(i,0) = 0, A(i > 0, j > 0) = A003961(A(i-1,j)).
%F A242378 For j > 0, A(i,j) = A297845(A000040(i+1),j) = A297845(j,A000040(i+1)). - _Peter Munn_, Sep 02 2025
%e A242378 The top-left corner of the array:
%e A242378 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
%e A242378 0, 1, 3, 5, 9, 7, 15, 11, 27, ...
%e A242378 0, 1, 5, 7, 25, 11, 35, 13, 125, ...
%e A242378 0, 1, 7, 11, 49, 13, 77, 17, 343, ...
%e A242378 0, 1, 11, 13, 121, 17, 143, 19,1331, ...
%e A242378 0, 1, 13, 17, 169, 19, 221, 23,2197, ...
%e A242378 ...
%e A242378 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.
%o A242378 (Scheme, with function factor from with Aubrey Jaffer's SLIB Scheme library)
%o A242378 (require 'factor)
%o A242378 (define (ifactor n) (cond ((< n 2) (list)) (else (sort (factor n) <))))
%o A242378 (define (A242378 n) (A242378bi (A002262 n) (A025581 n)))
%o A242378 (define (A242378bi row col) (if (zero? col) col (apply * (map A000040 (map (lambda (k) (+ k row)) (map A049084 (ifactor col)))))))
%Y A242378 Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.
%Y A242378 Transpose: A242379.
%Y A242378 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).
%Y A242378 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).
%Y A242378 Excluding column 0, a subtable of A297845.
%Y A242378 Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.
%K A242378 nonn,tabl,changed
%O A242378 0,4
%A A242378 _Antti Karttunen_, May 12 2014