cp's OEIS Frontend

A163263 Numbers having multiple representations as the product of non-overlapping ranges of consecutive numbers.

%I A163263 #9 Jul 05 2024 18:02:48
%S A163263 210,720,175560,17297280
%N A163263 Numbers having multiple representations as the product of non-overlapping ranges of consecutive numbers.
%C A163263 A subsequence of A064224. This sequence gives solutions P to the equation P = (x+1)...(x+m) = (y+1)...(y+n) with x>0, y>0 and x+m < y+1. So far, no numbers P with more than two representations have been discovered. Note that the only the lowest range of consecutive numbers (x+1 to x+m) can contain prime numbers; the other ranges are in a gap between consecutive primes. Gaps between the first 45000 primes were searched for additional terms, but none were found.
%H A163263 Art Kalb, <a href="https://www.youtube.com/watch?v=i8bn0US_k84">Why Number Theory is Hard</a>, YouTube video, 2024
%H A163263 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_469.htm">Puzzle 469. 5040</a>, The Prime Puzzles and Problems Connection.
%e A163263 210 = 5*6*7 = 14*15.
%e A163263 720 = 2*3*4*5*6 = 8*9*10.
%e A163263 175560 = 19*20*21*22 = 55*56*57.
%e A163263 17297280 = 8*9*10*11*12*13*14 = 63*64*65*66.
%o A163263 (Python)
%o A163263 import heapq
%o A163263 def aupton(terms, verbose=False):
%o A163263     p = 2*3; h = [(p, 2, 3)]; nextcount = 4; alst = []; oldv = None
%o A163263     while len(alst) < terms:
%o A163263         (v, s, l) = heapq.heappop(h)
%o A163263         if v == oldv and ((s > oldl) or (olds > l)) and v not in alst:
%o A163263             alst.append(v)
%o A163263             if verbose: print(f"{v}, [= Prod_{{i = {s}..{l}}} i = Prod_{{i = {olds}..{oldl}}} i]")
%o A163263         if v >= p:
%o A163263             p *= nextcount
%o A163263             heapq.heappush(h, (p, 2, nextcount))
%o A163263             nextcount += 1
%o A163263         oldv, olds, oldl = v, s, l
%o A163263         v //= s; s += 1; l += 1; v *= l
%o A163263         heapq.heappush(h, (v, s, l))
%o A163263     return alst
%o A163263 print(aupton(4, verbose=True)) # _Michael S. Branicky_, Jun 24 2021
%Y A163263 Cf. A064224.
%K A163263 nonn
%O A163263 1,1
%A A163263 _T. D. Noe_, Jul 29 2009