OFFSET
1,1
COMMENTS
Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ].
For prime p>3, p^2 divides H((p-1)/2,2p), implying that a(p)<=p. a(p)=p for prime p in {5,7,11,17,23,29,41,53,59,83,89,101,113,131,...}.
Note that many a(n) are of the form 2^m - 1 (for example, a(1) = 7, a(2) = 31, a(3) = 127, a(6) = 8191, etc.). a(n) = 5 for n = 5 + 10k, where k = {1,2,3,4,5,6,7,...}. a(n) = 7 for n = 1 + 3k, where k = {1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,17,19,20,...}. a(n) = 31 for n = 2 + 5k, where k = {2,6,8,9,12,14,...}.
a(50) > 3*10^6.
a(51)-a(62) = {17,7,53,131,5,7,19,7,59,23,7,31}. a(64)-a(77) = {7,5,11,7,17,23,7,23,31,7,37,5,7,7}. a(79)-a(119) = {7,47,263,7,83,2543,5,43,29,7,89,103,7,23,23,7,5,16193,7,7,11,7,101,17,7,13,5,7,31,127,7,37,37,7,113,19,5,29,13,7,7}. a(121)-a(150) = {7,31,41,7,5,23,7,37,43,7,131,11,7,67,5,7,23,23,7,7,47,7,11,1847,5,37,31,7,47,127}.
Currently a(n) is unknown for n = {50,63,78,120,...}.
LINKS
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
CROSSREFS
KEYWORD
hard,more,nonn
AUTHOR
Alexander Adamchuk, Jan 13 2007
EXTENSIONS
a(48), a(84), a(96), a(144) from Max Alekseyev, Sep 12 2009
STATUS
approved