Problems & Puzzles: Puzzles

Puzzle 428. Runs of consecutive integers such that... (II) As a follow-up for the Puzzle 427, I ask now for runs of r consecutive integers n, n+1, ...n+r-1 having f prime factors -repetitions allowed. Several people has worked out this subject before, as you will learn below by the links to published sequences. The records are as follows: f Largest known or possible run, r Smallest n 2 3 (largest possible) 33 3 7 (largest possible) 211673, A067813 4 15 (largest possible) 97524222465 A067814 5 12 393370860205 A067820 6 6 4843161124 A067821 7 6 242576758750 A067822 8 3 40909374 9 3 668363967 Example, f=5: 1       393,370,860,205 5*17^2*1051*259019 2       393,370,860,206 2*53*461*953*8447 3       393,370,860,207 3*13*463*3251*6701 4       393,370,860,208 2^4*24585678763 5       393,370,860,209 29*179*197*199*1933 6       393,370,860,210 2*3*5*21163*619589 7       393,370,860,211 7^2*167*673*71429 8       393,370,860,212 2^2*11*383*23342681 9       393,370,860,213 3^4*4856430373 10       393,370,860,214 2*131^2*977*11731 11       393,370,860,215 5*19*113*439*83471 12       393,370,860,216 2^3*3*16390452509 Evidently, the largest possible run is 2^f-1. Q1. Get larger runs for each f>4.

---