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All terms are primes. Note a connection to the Wieferich primes A001220: a(2) = a(3) = A001220(1), a(3) = a(4) = A001220(2).

From John Blythe Dobson, Jan 14 2017: (Start)

All Wieferich primes p will belong to this sequence twice, because if H([p/k]) denotes the harmonic number with index floor(p/k), then p divides all of H([p/4]), H([p/2]), and H(p-1). The first two of these elements gives one solution, and the second and third another. This property of the Wieferich primes predates their name, and was apparently first proved by Glaisher in "On the residues of r^(p-1) to modulus p^2, p^3, etc.," pp. 21-22, 23 (see References).

Note also a connection to the Mirimanoff primes A014127: a(1) = A014127(1), a(8) = A014127(2). All Mirimanoff primes p will belong to this sequence, because p divides both H([p/3]) and H([2p/3]). This property of the Mirimanoff primes likewise predates their name, and was apparently first proved by Glaisher in "A general congruence theorem relating to the Bernoullian function," p. 50 (see Links).

The Wieferich primes and Mirimanoff primes would seem to be the only cases for which the value of n in A126196(n) is predictable from knowledge of p. It is not obvious that all members of the present sequence are prime; however, by definition all their divisors must be non-harmonic primes A092102. Furthermore, it is clear from the cited literature under that entry that H([n/2]) == H(n) == 0 (mod p) is only possible when n < p. Thus, all divisors of the present sequence must belong to the harmonic irregular primes A092194.

One possible reason for interest in this sequence is a 1995 result of Dilcher and Skula (see Links) which among other things shows that if a prime p were an exception to the first case of Fermat's Last Theorem, then p would divide both H([p/k]) and H([2p/k]) for every value of k from 2 to 46. To date, the only values for which such coincidences have been found have k = 2, 3, or 4. For k = 6 to hold, p would have to be simultaneously a Wieferich prime and a Mirimanoff prime, while for k = 5 to hold, p would have to be simultaneously a Wall-Sun-Sun prime and a member of A123692. The sparse numerical results for the present sequence suggest that even the more relaxed condition H([n/2]) == H(n) == 0 (mod p) is rarely satisfied. (End)

Note a connection to the Wieferich primes A001220: a(2) = (A001220(1) - 1)/2, a(3) = A001220(1) - 1, a(4) = (A001220(2) - 1)/2, a(5) = A001220(2) - 1. [Comment regarding a(2) added by Kevin J. Gomez, Jul 11 2017]

a(9) > 840000. - Giovanni Resta, May 13 2016

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Sequence contains all geometric progressions of the form (p-1)*p^k for k > 0 and some primes p > 3. Note the factorization of initial terms of {a(n)} = {4*5, 6*7, 4*5^2, 10*11, 12*13, 16*17, 6*7^2, 18*19, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, 46*47, 4*5^4, 52*53, 58*59, 60*61, 66*67, 16*17^2, 70*71, 72*73, 78*79, 18*19^2, 82*83, ...}. The smallest term that does not fit this pattern is 11026 = ((149-1)/2) * 149.

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Note that {a(n)} contains the following geometric progressions: ((16843-1)/3)*16843^m found by Max Alekseyev, ((16843-1)/2)*16843^m found by Max Alekseyev, ((16843-1)*2/3)*16843^m, (16843-1)*16843^m, 20826*21647^m found by Max Alekseyev, ((2124679-1)/3)*2124679^m, ((2124679-1)/2)*2124679^m, ((2124679-1)*2/3)*2124679^m, (2124679-1)*2124679^m. Here {16843, 2124679} = A088164 are the only two currently known Wolstenholme Primes: primes p such that {2p-1} choose {p-1} == 1 mod p^4. See more details in Comments at A128672 and A125581.

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Sequence contains all terms of geometric progressions of the form (p-1)*p^k, k > 0, for some primes p >= 5, such as 4*5^k, 7*23^k, 15*23^k, 27*113^k, etc. Note the factorization of initial terms of {a(n)} = {4*5, 4*5^2, 10*11, 12*13, 7*23, 16*17, 18*19, 15*23, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 46*47, 4*5^4, 52*53, 27*113, 58*59, 60*61, 7*23^2, ...}. See more details in Comments at A128672 and A125581.

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Sequence contains all terms of geometric progressions 37^k*(37-1)/3, 37^k*(37-1)/2, 37^k*(37-1)*2/3, 37^k*(37-1) for k > 0. Note the factorization of initial terms of {a(n)} = {37*12, 37*18, 37*24, 37*36, ...}. See more details in Comments at A128672 and A125581.

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m}1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Some apparent periodicity in {a(n)} (not without exceptions): a(n) = 20 for n = 2 + 4m, a(n) = 42 for n = 4 + 12m and 8 + 12m, a(n) = 76 for n = 9 + 18m, a(n) = 77 for n = 1 + 10m, a(n) = 104 for n = 7 + 12m, a(n) = 110 for n = 12m, a(n) = 136 for n = 25 + 32m, etc.

See more details in Comments at A128672 and A125581.

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Sequence contains geometric progressions of the form (p-1)*p^k for k > 0 and some prime p > 5. Note the factorization of initial terms of {a(n)} = {6*7, 10*11, 12*13, 16*17, 6*7^2, 18*19, 22*23, 28*29, 30*31, 10*11*2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, ...}. See more details in Comments at A128672 and A125581.

All listed terms are composite.

The ratio of A117731(n) and A082687(n) when they are different is listed in A125741(n) = A117731[ a(n) ] / A082687[ a(n) ] = {7, 13, 7, 7, 37, 19, 119, 41, 31, 37, 37, 43, 13, 7, 13, 49, 7, 7, 61, 71, 103, 67, 73, 139, ...}.

It appears that all (or almost all) members of geometric progressions 2*7^k, 4*13^k, 15*7^k, 3^37^k, 6*19^k, 17*7^k, 4*41^k, 10*31^k, 12*37^k, 55*13^k, 107*7^k, etc. for k>0 are in the sequence.

Generalized harmonic numbers are defined as H(m,k) = Sum_{i=1..m} 1/i^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{i=1..m} (-1)^(i+1)*1/i^k.

a(18)..a(24) = {77,104,77,136,104,370,136}. a(26)..a(27) = {77,104}.

a(n) is currently unknown for n = {16,17,25,...}. See more details in Comments at A128672 and A125581.