2014-12-05, 17:32 | #1 |
Jan 2005
Minsk, Belarus
2^{4}×5^{2} Posts |
Consecutive p-smooth integers
Let's define L(n, k) as the largest prime factor of product
n*...*(n+k) of k+1 consecutive integers, starting at positive integer n. So we have, for example, L(4374, 1) = 7 L(48, 2) = 7 L(350, 2) = 13 L(138982582998, 2) = 103 L(61011223, 3) = 163 L(23931257472314, 3) = 631 L(1517, 4) = 41 L(3294850, 5) = 239 L(1913253200, 8) = 3499 L(8559986129664, 12) = 58393 L(48503, 14) = 379 Conjecture: as n goes to infinity, lim inf L(n, k) / (log n)^2 = C_k The very rough estimates of constants C_k are: C_1 ~ 0.0376 C_2 ~ 0.258 C_3 ~ 0.907 C_4 ~ 2.46 C_5 ~ 5.16 C_6 ~ 11.4 C_7 ~ 19 C_8 ~ 42 C_9 ~ 70 C_10 ~ 140 C_11 ~ 200 C_12 ~ 250 C_13 ~ 380 C_14 ~ 430 C_15 ~ 460 Some successive minima of L(n, k) are shown there: http://oeis.org/A193943 http://oeis.org/A193944 http://oeis.org/A193945 http://oeis.org/A193946 http://oeis.org/A193947 http://oeis.org/A193948 http://oeis.org/A199407 http://oeis.org/A200566 http://oeis.org/A200567 http://oeis.org/A200568 http://oeis.org/A200569 http://oeis.org/A200570 http://oeis.org/A209837 http://oeis.org/A209838 http://oeis.org/A209839 Any suggestions on the conjecture? Does it depend on other known ones like Twin prime conjecture or ABC conjecture? Great thanks for any information on the subject. |
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