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

Author

Reinhard Zumkeller, Aug 22 2005

Keywords

• nonn
• base
• frac

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

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

Extensions

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