Harmonic-Number-Related Divisibility

[Ralf Stephan]

> Could someone please (also) submit, if it is not already in the EIS, the 
> above sequence divided by the appropriate odd primes?

That was what I did (before you gave your proof so that appears as
proposition) and I noticed something curious: apart from the a(n)
being even, they have very few small prime factors other than 2 (see
below). Now one may handwave and say that's because they cancel out
in H(n) but then what does the single 7 in there?

Mat([0, 1])
Mat([2, 1])
[2, 1; 31, 1]
[2, 1; 277, 1]
[2, 1; 107, 1; 347, 1]
[2, 1; 71341, 1]
[2, 1; 1415473, 1]
[2, 1; 2411, 1; 300493, 1]
[2, 1; 6288207713, 1]
[2, 1; 181, 1; 937, 1; 5477341, 1]
[2, 1; 587, 1; 55054614499, 1]
[2, 1; 1277, 1; 9769, 1; 2515907, 1]
[2, 1; 7, 1; 617, 1; 19043510162059, 1]
[2, 1; 246781, 1; 710634722945863, 1]
[2, 1; 17903, 1; 161276144488536257, 1]
[2, 1; 2990527, 1; 944275513155287, 1]
[2, 1; 102470147891, 1; 62067905422387, 1]
[2, 1; 136526172581730681305097341, 1]
[2, 1; 2243, 1; 179147077171799688729907, 1]
[2, 1; 20219, 1; 2647687177, 1; 110490689429683529, 1]
[2, 1; 1621, 1; 2377, 1; 2843, 1; 3739, 1; 1001902316007466451, 1]
[2, 1; 191, 1; 65587, 1; 56755058929, 1; 41043040334400947, 1]
[2, 1; 4149373, 1; 15034559, 1; 430674677178974312684359, 1]
[2, 1; 13457321, 1; 387499171, 1; 517687273, 1; 937745784573449, 1]
[2, 1; 191, 1; 4951, 1; 930368089977257, 1; 113479910817830873, 1]
[2, 1; 191, 1; 93275951618777, 1; 56712880614928739514061241, 1]
[2, 1; 2097653, 1; 2376195853996077517463363660291640073, 1]
[2, 1; 751, 1; 1621, 1; 14020824120169849, 1; 301605617848055454607351, 1]
[2, 1; 313, 1; 41280594376615095973857246066208675722839798171, 1]
[2, 1; 977, 1; 8677, 1; 47294231320083408073116552242426346713203976099, 1]
[2, 1; 3307, 1; 913216253101, 1; 16822731118285155147477806842977762991409, 1]
[2, 1; 45763, 1; 307382187793, 1; 3572934567923652856125597952630887958231, 1]
[2, 1; 181, 1; 3587528314036749155797375501, 1; 1399360439401613862728338546057, 1]
[2, 1; 1932541510450309679, 1; 274723321296069709307, 1; 1694474294473682876111, 1]
[2, 1; 631, 1; 236713, 1; 10666438761703, 1; 12337496050590977477997516084765696987186379, 1]
[2, 1; 3917, 1; 58367, 1; 3431677134278546715028988413, 1; 294129855515822178580364578783, 1]
[2, 1; 653, 1; 1987, 1; 55969334471, 1; 18650089070602528737707, 1; 9089227057040294922736855367, 1]
[2, 1; 8161, 1; 1126828312451, 1; 46028100811088177, 1; 799143102018431503045140869803831957379, 1]
[2, 1; 311569, 1; 171253083707, 1; 1068669625813610558537914767269088872058123090632156419001, 1]
[2, 1; 761, 1; 2038210961490049, 1; 147636598206060117062379671, 1; 246906478668841826947295692763, 1]
[2, 1; 313, 1; 1783, 1; 143113, 1; 162627539, 1; 701595776621, 1; 13034885953480103, 1; 14801084676288014718541111615807, 1]

Regards,
ralf