1,1

Generalized harmonic numbers are defined as H(n,k) = Sum[ 1/i^k, {i,1,n} ]. Alternating generalized harmonic numbers are defined as H'(n,k) = Sum[ (-1)^(i+1)*1/i^k, {i,1,n} ]. a(3) = A128673(1) = 94556602, a(13) = 3504, a(17) = 1107, a(23) = 2948 and a(37) = 1752 found by Max Alekseyev. a(36)-a(38) = {110,1752,20}. a(40)-a(46) = {42,77,20,104,42,76,20}. a(48) = 110. a(50)-a(52) = {20,77,42}. a(54)-a(58) = {20,104,42,136,20}. a(60)-a(64) = {272,77,20,76,42}. a(66)-a(74) = {20,104,42,556,20,77,110,136,20}. a(76)-a(82) = {42,424,20,104,42,76,20}. a(84) = 110. a(86) = 20. a(88)-a(92) = {42,136,20,77,42}. a(94) = 20. a(96) = 110. a(98)-a(104) = {20,76,42,77,20,104,42}. a(n) is currently unknown for n = {35,39,47,49,53,59,65,75,83,85,87,93,95,97,...}. Note some apparent periodicity in a(n) (not without exclusions): 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 for A128672 and A125581.

Eric Weisstein, Link to a section of The World of Mathematics. Harmonic Number.

Cf. A001008, A002805, A058313, A058312. Cf. A007406, A007407, A119682, A007410, A120296. Cf. A125581 = numbers n such that n does not divide the denominator of the n-th harmonic number nor the denominator of the n-th alternating harmonic number. Cf. A126196, A126197. Cf. A128672 = numbers n such that n^k does not divide the denominator of the n-th generalized harmonic number H(n, k) nor the denominator of the n-th alternating generalized harmonic number H'(n, k), for k = 2. Cf. A128673, A128674, A128675, A128676, A128671 = A128670(Prime(n)).

Sequence in context: A116255 A136609 A116246 this_sequence A033397 A052202 A089525

Adjacent sequences: A128667 A128668 A128669 this_sequence A128671 A128672 A128673

hard,more,nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Mar 24 2007