I understand or recollect that shi-is the English spelling of the Japanese six. shi-shi is the spelling of seven
Pretty close, but not quite.
Four - SHI or YON (preferred)
Five - GO
Six - ROKU
Seven - SHICHI or NANA (preferred)
SHI and SHICHI are not used as often as the other reading. Although you would use those readings for somethings, like the words for April (SHIGATSU) and July (SHICHIGATSU).