From Jon E. Schoenfield, Aug 27 2019, updated Apr 20 2023: OEIS sequence A309881 is "a(n) is the smallest number k such that the value of n can be deduced given only the values tau(n), tau(n+1), ..., tau(n+k), where tau is the number of divisors function. I.e., A309981(n) is the smallest k such that there exists no number m != n for which tau(m+j) = tau(n+j) for all j in 0..k, where tau is the number of divisors function A000005. Let m(n) be the smallest nonnegative integer such that tau(m+j) = tau(n+j) for all j in 0..A309981(n)-1. In other words, although the vector of tau values (tau(n), tau(n+1), ..., tau(n+A309981(n)) uniquely identifies n, removing the last element of that vector leaves (tau(n), tau(n+1), ..., tau(n+A309981(n)-1), which is equal to the vector (tau(m(n)), tau(m(n)+1), ..., tau(m(n)+A309981(n)-1). The table below gives m(n) for n = 1..87. Example: A309981(19) = 4, so the vector (tau(19), tau(20), tau(21), tau(22), tau(23)) = (2, 6, 4, 4, 2) is the vector (tau(n), tau(n+1), tau(n+2), tau(n+3), tau(n+4)) only for n = 19. From the table below, m(19) = 31, so the vectors (tau(19), tau(20), ..., tau(23)) = (2, 6, 4, 4, 2) and (tau(31), tau(32), ..., tau(35)) = (2, 6, 4, 4, 4) do not differ until their 5th element, i.e., tau(19+4) != tau(31+4). ============================================================= n m(n) ------ 1 2 2 3 3 2 4 9 5 7 6 10 7 5 8 6 9 25 10 6 11 17 12 18 13 37 14 21 15 6 16 81 17 11 18 12 19 31 20 716 21 57 22 6 23 29 24 30 25 121 26 1226 27 51 28 6196 29 41 30 66 31 211 32 92 33 85 34 14 35 6 36 100 37 1381 38 86 39 1191 40 136 41 29 42 906 43 331 44 9836 45 261 46 6 47 79 48 80 49 1681 50 722 51 15812931 52 594532 53 27302213 54 540534 55 147415 56 23096 57 15681 58 106 59 71 60 72 61 73 62 27 63 12 64 729 65 785 66 61986 67 61987 68 2641564 69 185529 70 25930 71 59 72 60 73 72970801 74 34017362 75 983403 76 112204 77 365 78 366 79 47 80 48 81 625 82 670762 83 18803 84 337044 85 186565 86 38 87 3814791