Some Beacon Paperbacks were published under the Ariadne Book imprint. The back cover of one of these titles provides a description:
Ariadne Books are limited edition paperbacks for which there is a small but steady demand. Instead of putting such titles out of print, Beacon Press is publishing them under this special imprint. To keep the price down, Ariadne Books will be available only in paperbound editions.
Beacon to Issue Paperback Reprints and Originals, PW, Jan. 15, 1955, at 210 (announcing April 20, 1955, launch of Beacon's new paperbound books series in "full library size" priced from $1.25 to $1.95, with reprints and some originals, to be published in both paper and in cloth. Noting also that Beacon hopes "the books may help serve 'as a counteroffensive to Soviet books.'")
Ad, PW, Jan. 29, 1955, at 429 (showing first 6 titles, to be published in April 1955).
PW, Oct. 7, 1957, at 74 (note that BP 51, The Intimate Journals of Charles Baudelaire, is the first book ever printed on Whiteford's Impact Book paper, a type of tinted paper intended to be easier on the eyes of readers).
Beacon published a few paperbacks with special series numbers. I'm guessing the "S" in BP S means "Special," as Peace: The Control of National Power is a guide for concerned citizens. The Unitarian Universalist Association announced in The Leader (can't verify which issue, but I think it's 1965) that "[t]his spring, four books by A. Powell Davies are being republished in paperback by the Beacon Press for the Davies Memorial committee, in time for the UUA General Assembly." These four books were given designations BP A-D. Finally, BP 5000 is an out-of-sequence "tract" (as PW called it) about the right-to-life movement.