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

1,2

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.

Formula

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

Mathematica

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

Prog

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

Crossrefs

Cf. A070939, A359451.

Sequence in context: A079433 A319205 A110728 * A190424 A322484 A066332

Adjacent sequences: A359447 A359448 A359449 * A359451 A359452 A359453

Keyword

nonn,base

Author

Amiram Eldar, Jan 02 2023

Status

approved