Get list of skus and their corresponding category_ids with direct SQL Magento.

skus-with-category_ids.sql

SELECT sku, GROUP_CONCAT(entity_id) as category_ids FROM (
  SELECT c3.sku, c2.entity_id
  FROM catalog_category_product c1
  INNER JOIN catalog_category_entity_varchar c2 ON (c1.category_id = c2.entity_id)
  INNER JOIN catalog_product_entity c3 ON (c1.product_id = c3.entity_id)
  WHERE c2.attribute_id = (SELECT attribute_id FROM eav_attribute WHERE attribute_code = 'name' AND entity_type_id = 3)
) main
GROUP BY sku