CHAP, xxj PROPERTIES OF THE DIGITS OF NUMBERS. 457
Moret-Blanc23 proved that 1, 8, 17, 18, 26, 27 are the only numbers equal | to the sum of the digits of their cubes.
C. Berdelle*230 considered the last n digits of numbers, in particular of 5*.
E. Ces&ro24 noted that the sum of the pth powers of ten consecutive integers ends with 5 unless p is a multiple of 4, when it ends with 3. wt F. de Rocquigny25 noted that if a number of n digits equals the sum of the 2n~l products of its digits taken 1, 2, . . ., n at a time, its final n—1 digits are all 9.
E. Ces£ro26 considered the period of the digits of rank n in powers of 5.
Lists2*80 have been given of squares formed by the nine digits > 0, or the ten digits, not repeated.
0. Kessler27 gave a table of divisors of numbers formed by repeating a given set of digits a small number of times.
T. C. Simmons270 noted that, if the sum of the digits of n is 10, that of 2n is 11 unless each digit of n is <5 or two are 5. For 4 digits the numbers of each type are counted.
J. S. Mackay28 treated the last subject.
E. Lemoine29 considered numbers like A = 8607004053 such that, if a is the number derived by reversing the digits of A, the sum A +a = 12111011121 reverses into itself. /
M. d'Ocagne30 considered the sum a"(N) of the digits of the first N integers. If Nf = a9-lW+ . . . +arlO+ao and d=a3,-10p~l, then
Hence
The number of digits in 1,. . ., N is next paper.
M. d'Ocagne31 noted that, in writing down the natural numbers 1, where N is composed of n digits, the total number of digits wi n(tf +1) - In, where ln = 1 . . . 1 (to n digits).
E. Barbier310 asked what is the 101000th digit written if the series of natural numbers be written down.
MNouv. Ann. Math., (2), 18, 1879, 329; proposed by Laisant, 17, 1878, 480.
»°Assoc. frang., 8, 1879, 176-9.
S4Nouv. Corresp. Math., 6, 1880, 519; Mathesis, 1888, 103.
"Lea Mondes, 53, 1880, 410-2.
"Nouv. Corresp. Math., 4, 1878, 387; Nouv. Ann. Math., (3), 2, 1883, 144, 287; 1884, 160.
MflMath. Magazine, 1, 1882-4; 69-70; 1'interme'diaire des math., 4, 1897, 168; 14, 1907, 135;
Sphinx-Oedipe, 1908-9, 35; 5, 1910, 64; Educ. Times, March, 1905. Math. Quest. Educ.
Times, 52, 1890, 61; (2), 8, 1905, 83-6 (with history). "Zeitschrift Math. Phys., 28, 1883, 60-64. 27°Math. 'Quest. Educ. Times, 41, 1884, 28-9, 64-5. "Proc. Edinburgh Math. Soc., 4, 1885-6, 55-56. "Nouv. Ann. Math., (3), 4, 1885, 150-1. "Jornal de sc. math, e ast., 7, 1886, 117-128. "Ibid., 8, 1887, 101-3; Comptes Rendus Paris, 106, 1888, 190. 31flComptes Rendus Paris, 105, 1887, 795, 1238.