A359450 a(1) = 1, a(2) = 2; thereafter a(n) = n * a(A070939(n)).

1, 2, 6, 24, 30, 36, 42, 192, 216, 240, 264, 288, 312, 336, 360, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 1152, 1188, 1224, 1260, 1296, 1332, 1368, 1404, 1440, 1476, 1512, 1548, 1584, 1620, 1656, 1692, 1728, 1764, 1800, 1836

Offset: 1

Author

Amiram Eldar, Jan 02 2023

Keywords

• nonn
• base

Comments

Problem A6 of the 63rd Putnam Competition (2002) asked to prove that when this sequence is generalized to base-b digits, the sum of reciprocals converges only for b = 2.

Problem 2 in Appendix D of Bornemann et al. (2004) asked to calculate the sum of the reciprocals of this sequence.

References

• Daniel D. Bonar and Michael J. Khoury, Jr., Real infinite Series, The Mathematical Association of America, 2006, pp. 159, 190-191.
• Hongwei Chen, Classical Analysis: An Approach through Problems, CRC Press, 2022, p. 118, exercise 34.
• Kiran S. Kedlaya, Daniel M. Kane, Jonathan M. Kane, and Evan M. O'Dorney, The William Lowell Putnam Mathematical Competition 2001-2016: Problems, Solutions, and Commentary, American Mathematical Society, 2020, pp. 86-87.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Folkmar Bornemann, Dirk Laurie, Stan Wagon, and Jörg Waldvogel, The SIAM 100-Digit Challenge, A Study in High-Accuracy Numerical Computing, SIAM, Philadelphia, 2004. See Appendix D, Problem 2, p. 281.
• Kiran Kedlaya and Lenny Ng, Solutions to the 63rd William Lowell Putnam Mathematical Competition, Saturday, December 7, 2002. See Problem A-6, p. 2.
• Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, The Sixty-Third William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 110, No. 8 (2003), pp. 718-726.
• John Scholes, Problem A6, The 63rd Putnam Competition, 2002.
• David Smith, On a slowly converging sum, Solution to Problem 2, 2003.

Crossrefs

Cf. A070939, A359451.

Programs

Mathematica

a[1] = 1; a[2] = 2; a[n_] := a[n] = n * a[BitLength[n]]; Array[a, 100]

PARI

a(n) = if(n < 3, n, n * a(#binary(n)));

Formula

Sum_{n>=1} 1/a(n) = A359451.