
COMMENTS

By definition a(1) is A045345(2).
This sequence has a very interesting behavior. If Mod(n, 2)(Mod(n, 20)1)(Mod(n, 20)9)(Mod(n, 20)13)(Mod(n, 20)17)!=0, a(n)=17, 23 or 25; in other cases a(n) may be too large. If Mod[n, 16] = 15, a(n) = 17. For example, a(n) = 17 for n = 15, 31, 47, 63, 79, 95, 111, 127, 143, 159, 175, 191, ...; also, a(n) = 23 for n = 1, 13, 23, 35, 45, 57, 67, 89, 101, 123, 133, 145, 155, 167, 177, 189, 199, ...; a(n) = 25 for n = 3, 5, 7, 11, 19, 25, 27, 39, 43, 51, 55, 59, 65, 71, 75, ..., . For a(n) = 19 for n = 2, 20, 38, 56, 74, 92, 110, 128, 146, 164, 182, 200, 218, ..., == 2 (mod 18).
From Alexander Adamchuk, Jul 20 2008: (Start)
Conjectures:
a(n) exists for all n; a(n) >= 17.
a(325)a(575) = {25,19,25,5851,1843,61,23,821,89,301,17,37,131,455,25,1607,297,37,23,19,25,
325,25,37,353,47,17,1663,23,691,25,691,509269,155,25,269,105893,19,25,3971,
23,213215,17,26021,327,79,25,37,151,83,23,161,101,37,25,19,327,265,17,37,25,
43,23,41,169,61,25,113,21761,6289,25,47,23,19,17,4073,1137,565,25,527,25,
325,25,37,23,455,25,431,13195,37,17,19,53,155,23,37,89,455,25,18839,25,6221,
25,41,18597,229,17,811,623173,19,25,193,2079,673,25,881,23,47,25,37,25,97,
17,79,131,37,25,19,23,56501,25,37,299,455,25,167,2707,446963,17,157,25,325,
25,41,53,19,25,5917,103,1051,23,607,101,155,17,37,6233,455,25,9049,23,37,25,
19,327,5359,25,37,43,455,17,9187,23,193,25,1861,7923,301,25,113,25,19,23,41,
89,61,17,43,1785,131,25,37,1417,455,23,151,53,37,25,19,25,79,17,37,23,455,
25,289,59,47,25,511,47,83,25,739,23,19,17,301,25,269,25,41,707,2735,23,37,
299,43,25,283,69723,37,17,19,1785,479,23,37,25,455,25,1867,131,61,25,31799,
23,161,17}.
a(n) is currently unknown and a(n)>10^7 for n = {324, 576, ...}. (End)
All but one of the terms up to n=1000 are known and they are less than 10^8. Currently the only unknown term for n<=1000 is a(656)>10^8.  Alexander Adamchuk, May 24 2009
a(656) > 23,491,000,000.  Robert Price, Apr 22 2014
