A337922 Numbers k such that when the first digit of k is shifted to the end the result is 3*k/2.

1176470588235294, 2352941176470588, 3529411764705882, 4705882352941176, 5882352941176470, 11764705882352941176470588235294, 23529411764705882352941176470588, 35294117647058823529411764705882, 47058823529411764705882352941176, 58823529411764705882352941176470

Offset

1,1

Comments

The problem of finding the least number in this sequence was suggested by the Polish-British mathematician and historian Jacob Bronowski (1908-1974).

Anderson (1988) credited the problem to the British mathematician John Edensor Littlewood (1885-1977). The solution to the problem was given in 1955 by the British mathematician Dudley Ernest Littlewood (1903-1979), a student of J. E. Littlewood (but they were not related).

References

Jacob Bronowski, New Statesman and Nation, Vol. 39, Dec. 24, 1949, p. 761.

Dan Pedoe, The Gentle Art of Mathematics, Macmillan, 1960, p. 11.

Links

Amiram Eldar, Table of n, a(n) for n = 1..310

Oliver D. Anderson, On Littlewood's Little Puzzle, Teaching Mathematics and its Applications: An International Journal of the IMA, Vol. 7, No. 3 (1988), pp. 144-146.

J. H. Clarke, Note 2298. A Digital Puzzle, The Mathematical Gazette, Vol. 36, No. 318 (1952), p. 276.

Keith Devlin, Micro-Maths, Mathematical problems and theorems to consider and solve on a computer, Macmillan, 1984, pp. 38-39.

D. E. Littlewood, Note 2494. On Note 2298: a digital puzzle, The Mathematical Gazette, Vol. 39, No. 327 (1955), p. 58.

Joseph S. Madachy, Recreational Mathematics, The Fibonacci Quarterly, Vo. 6, No. 6 (1968), pp. 385-398. See page 389.

Math Stackexchange, Finding the Dr. Bronowski's number, 2015.

Sidney Penner, Problem E 1530, Elementary Problems and Solutions, The American Mathematical Monthly, Vol. 69, No. 7 (1962), p. 667; J. W. Ellis and others, Placing the First Digit Last, Solution to Problem E 1530, ibid., Vol. 70, No. 4 (1963), pp. 441-442.

D. G. Rogers, Jacob Bronowski (1908-1974) The Mathematical Gazette and retrodigitisation, The Mathematical Gazette, Vol. 92, No. 525 (2008), pp. 476-479; alternative link.

Wikipedia, Transposable integer.

Formula

The decimal digits of the first 5 terms are the periodic parts of the decimal expansions of 2/17, 4/17, 6/17, 8/17 and 10/17. The next terms are all the concatenations of each of these terms with itself an integral number of times (Anderson, 1988).

Example

1176470588235294 is a term since 1764705882352941 = 3*1176470588235294/2.

Mathematica

concat[n_, m_] := NestList[FromDigits[Join[{#}, IntegerDigits[n]]] &, n, m]; s = Range[2, 10, 2]*(10^16 - 1)/17; Union @ Flatten[concat[#, 2] & /@ s]

Crossrefs

Cf. A061242, A166320, A258663.

Sequence in context: A095431 A362445 A282672 * A072719 A185433 A134692

Adjacent sequences: A337919 A337920 A337921 * A337923 A337924 A337925

Keyword

nonn,base

Author

Amiram Eldar, Jan 29 2021

Status

approved