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A056534 Mapping from the ordering by product (A027750, A056538) to the ordering by sum (A002260, A004736) of ordered pairs (a,b), a>=1, b>=1.

%I A056534 #16 Jul 02 2025 23:16:09
%S A056534 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 A056534 55,56,66,67,23,18,19,27,78,79,91,92,30,35,105,106,24,26,120,121,38,
%U A056534 25,44,136,137,153,154,47,31,34,54,171,172,190,191,57,32,33,65,210,211,39
%N A056534 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 A056534 Ivan Neretin, <a href="/A056534/b056534.txt">Table of n, a(n) for n = 1..10000</a>
%H A056534 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A056534 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 A056534 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 A056534 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 A056534 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 A056534 Inverse: A056535.
%K A056534 nonn
%O A056534 1,2
%A A056534 _Antti Karttunen_, Jun 20 2000