hihi, all - several people have reminded me that there are some sequences related to the divisor chains one of them must start with n one of them must start with a divisor of n*(n+1)/2 (so it can form a cycle) actually, if the requirement is that each cyclic permutation of the sequence is also a divisor chain, then there are none (except the trivial one 1 for n=1), since not all integers start divisor chains for any given n>1 all of these subsequent counts were done by manual editing of the sequence output from the first program, so typos are quie possible the first counts are when the sequence must start with n (i call these anchored): 1 chains of length 1 1 chains of length 2 @ 1 chains of length 3 * 1 chains of length 4 @ 1 chains of length 5 * 1 chains of length 6 @ 1 chains of length 7 * 5 chains of length 8 @ 4 chains of length 9 * 3 chains of length 10 @ 2 chains of length 11 8 chains of length 12 4 chains of length 13 * 6 chains of length 14 47 chains of length 15 44 chains of length 16 @ 6 chains of length 17 * 37 chains of length 18 @ 6 chains of length 19 166 chains of length 20 462 chains of length 21 232 chains of length 22 @ 372 chains of length 23 2130 chains of length 24 1589 chains of length 25 * 9093 chains of length 26 @ (* = n and (n+1)/2 are prime powers, n odd) (@ = n/2 and n+1 are prime powers, n even) the second case is when the first element must divide n*(n+1)/2 (i call these cyclic): 1 chains of length 1 1 chains of length 2 2 chains of length 3 1 chains of length 4 2 chains of length 5 0 chains of length 6 3 chains of length 7 0 chains of length 8 5 chains of length 9 0 chains of length 10 6 chains of length 11 0 chains of length 12 6 chains of length 13 0 chains of length 14 145 chains of length 15 0 chains of length 16 22 chains of length 17 2 chains of length 18 27 chains of length 19 165 chains of length 20 57 chains of length 21 0 chains of length 22 516 chains of length 23 2021 chains of length 24 1912 chains of length 25 453 chains of length 26 the third case is when both conditions must hold: 1 chains of length 1 1 chains of length 2 1 chains of length 3 1 chains of length 4 1 chains of length 5 0 chains of length 6 1 chains of length 7 0 chains of length 8 4 chains of length 9 0 chains of length 10 2 chains of length 11 0 chains of length 12 4 chains of length 13 0 chains of length 14 47 chains of length 15 0 chains of length 16 6 chains of length 17 0 chains of length 18 6 chains of length 19 0 chains of length 20 0 chains of length 21 0 chains of length 22 372 chains of length 23 0 chains of length 24 1589 chains of length 25 0 chains of length 26 more later, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal@aero.org