A111935 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.

1, 3, 11, 25, 137, 49, 363, 761, 789, 8959, 27647, 368651, 377231, 128413, 261831, 4531207, 41461543, 8414831, 8531519, 8642903, 201237217, 203585563, 5145999379, 5200191979, 15757132337, 15908097437, 16048998197, 501745966907

Offset

1,2

Comments

Denominator = A111936;

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

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.

References

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

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

Links

Table of n, a(n) for n=1..28.

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A129082 A190476 A060746 * A175441 A001008 A375523

Adjacent sequences: A111932 A111933 A111934 * A111936 A111937 A111938

Keyword

nonn,base,frac

Author

Reinhard Zumkeller, Aug 22 2005

Extensions

Definition edited by N. J. A. Sloane, Dec 30 2019

Status

approved