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

Author

Amiram Eldar, Jan 29 2021

Keywords

• nonn
• base

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).

Examples

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

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.

Crossrefs

Cf. A061242, A166320, A258663.

Programs

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]

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).