The BCR tree can be reduced using this argument. Here are some examples. with(numtheory): f:=x->evalf(sigma(x)/x): line 201-209 Suppose 3^2 || N, So 13^2 | N, then f(3^2*5^2*13^2*23^2)==2.027479036 Suppose 3^4 | N, then f(3^4*5^2*23^2*31^2)==2.000863016 line 210-222 Suppose 3^2 || N, So 13^2 | N, then f(3^2*5^2*13^2*31^2)==2.004069479 Suppose 3^4 || N, So 11^2 | N, then f(3^4*5^2*11^2)==2.036049383 Suppose 3^6 | N, then f(3^6*5^2*31^2*79^2*101^12*127^2*157^2*313^2*521^2)==2.003665280 // Very nice line 3542-3551 Suppose 3^2 || N, So 13^2 | N, then f(3^2*5*13^2*19^6*43^2)==2.028342414 Suppose 3^4 | N, then f(3^4*5*19^6*43^2*61^2*71^2*373^2)==2.002986609 line 4344-4358 f(3^2*7^4*11^2*31^2*41^2*467^4*2801*3409261^2)==1.966607090 Suppose 3^2 || N, So 13^2 | N, then f(3^2*7^4*11^2*13^2)==2.005637529 Suppose 3^4 | N, then f(3^4*7^4*11^2*31^2*41^2)==2.033841520 line 5083-5097 f(3^4*7^6*13^2*29^2*109^2*263^2*419^2*541^2*571^2*1279^2*4733)==1.993994362 Suppose 3^4 || N, So 11^2 | N, then f(3^4*7^6*11^2*13^2)==2.074327235 Suppose 3^6 | N, then f(3^6*7^6*13^2*29^2*109^2*263^2*419^2*541^2*571^2*1279^2)==2.000895738 // Nice line 7167-7216 f(3^2*5^2*11^4*179^2*3221)==1.981893351 Suppose 3^2 || N, So 13^2 | N, then f(3^2*5^2*11^4*13^2)==2.133422650 Suppose 3^4 | N, then f(3^4*5^2*11^4)==2.037567595 // The avoided subtree is quite large line 7218-7232 f(3^2*5^2*11^4*179^2*3221)==1.981893351 Suppose 3^2 || N, So 13 | N, then f(3^2*5^2*11^4*13^2)==2.133422650 Suppose 3^4 | N, then f(3^4*5^2*11^4)==2.037567595 line 9100-9117 13^2 => 3*61 61^4 => 5*131*21491 21491^2 => 421*1097113 1097113^1 => 548557 f(3^2*5^2*13^2*61^4*131^2*421^2*21491^2*548557^2*1097113)==1.9918078665839148480 Suppose 5^2 || N, So 31^2 | N, then f(3^2*5^2*13^2*31^2)==2.0040694789081885856 Suppose 5^4 | N, then f(3^2*5^4*13^2*61^4*131^2)==2.0023619261323686327 line 9117-9130 13^2 => 3*61 61^4 => 5*131*21491 21491^2 => 421*1097113 1097113^2 => 3*37*5743*1888171 f(3^2*5*13^2*37^2*61^4*131^2*421^2*5743^2*21491^2*1097113^2*1888171^2)==1.9814025604942501611 Suppose 5 || N, So 3^4 | N, then f(3^4*5*13^2*37*61^4)==2.0267792028410419716 Suppose 5^2 || N, So 31^2 | N, then f(3^2*5^2*13^2*31^2)==2.0040694789081885856 Suppose 5^4 | N, then f(3^2*5^4*13^2*37)==2.0073269577269577270 // line 2940-2975 f(3^4*7^2*13^2*19^6*331^2*409^2*701^2*2017^2)==2.000954140