%I #16 Jul 02 2025 23:16:09

%S 1,2,3,4,6,7,5,10,11,15,16,8,9,21,22,28,29,12,14,36,37,13,45,46,17,20,

%T 55,56,66,67,23,18,19,27,78,79,91,92,30,35,105,106,24,26,120,121,38,

%U 25,44,136,137,153,154,47,31,34,54,171,172,190,191,57,32,33,65,210,211,39

%N Mapping from the ordering by product (A027750, A056538) to the ordering by sum (A002260, A004736) of ordered pairs (a,b), a>=1, b>=1.

%H Ivan Neretin, <a href="/A056534/b056534.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e The "ordering by sum": (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),(1,4),(2,3),(3,2),(4,1),...

%e The "ordering by product": (1,1),(1,2),(2,1),(1,3),(3,1),(1,4),(2,2),(4,1),(1,5),(5,1),...

%p ordered_pair_perm := proc(upto_n) local a,i,j; a := []; for i from 1 to upto_n do for j in sort(divisors(i)) do a := [op(a),binomial(((i/j) + j - 1),2)+j]; od; od; RETURN(a); end;

%t max = 21; A056534 = {}; For[i = 1, i <= max, i++, Do[ AppendTo[ A056534, Binomial[i/j + j - 1, 2] + j], {j, Divisors[i]}]]; A056534 (* _Jean-François Alcover_, Oct 05 2012, after Maple *)

%Y Inverse: A056535.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jun 20 2000