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Numerator of n-th term of the harmonic series after removal of all terms 1/m from Sum_{m=1..n} 1/m for which m contains a 9 in its decimal representation.
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%I #23 Sep 08 2022 08:45:21

%S 1,3,11,25,137,49,363,761,789,8959,27647,368651,377231,128413,261831,

%T 4531207,41461543,8414831,8531519,8642903,201237217,203585563,

%U 5145999379,5200191979,15757132337,15908097437,16048998197,501745966907

%N Numerator of n-th term of the harmonic series after removal of all terms 1/m from Sum_{m=1..n} 1/m for which m contains a 9 in its decimal representation.

%C Denominator = A111936;

%C Lim_{n->infinity} a(n)/A111936(n) = C < 80.

%C The sum of the harmonic series after removing all terms containing a 9 in decimal representation in decimal system converges and the sum is < 80. Hence the sum of the harmonic series in which at least one digit is missing (from 0 to 9) converges and the sum is less than 810.

%D G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 3, sect. 4, Problem 124.

%D Jason Earls and Amarnath Murthy, Some fascinating variations in harmonic series, Octogon Mathematical Magazine, Vol. 12, No. 2, 2004.

%e n=9: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/10 = 789/280, therefore a(9) = 789.

%o (Magma) a:=[k:k in [1..100]| not 9 in Intseq(k)]; [Numerator( &+[1/a[m]: m in [1..n]]): n in [1..30] ]; // _Marius A. Burtea_, Dec 30 2019

%Y Cf. A001008, A007095, A082838, A111936 (denominators).

%K nonn,base,frac

%O 1,2

%A _Reinhard Zumkeller_, Aug 22 2005

%E Definition edited by _N. J. A. Sloane_, Dec 30 2019