Chap. XX] PROPERTIES OF THE DiGITS OF NuMBERS. 457 
Moret-Blanc^^ proved that 1, 8, 17, 18, 26, 27 are the only numbers equal 
to the sum of the digits of their cubes. 
C. Berdelle^^" considered the last n digits of numbers, in particular of 5*. 
E. Cesaro^^ 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. 
F. de Rocquigny^^ noted that if a number of n digits equals the sum of the 
2" — 1 products of its digits taken 1, 2, . . ., n at a time, its final n — 1 digits 
are all 9. 
E. Cesaro^® considered the period of the digits of rank n in powers of 5. 
Lists^" have been given of squares formed by the nine digits > 0, or the 
ten digits, not repeated. 
0. Kessler^^ gave a table of divisors of numbers formed by repeating a 
given set of digits a small number of times. 
T. C. Simmons"" 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. Mackay^^ treated the last subject. 
E. Lemoine^^ 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'Ocagne^° considered the sum a{N) of the digits of the first N integers. 
If iVp = ap-10^+ . . . +ai-10+ao and d = ap-10^-l, then 
aid) = 10'-'-5a^{a,-l+9p), (7{N,) =(T{d) + {N,., + l)a^+a{N,.,). 
Hence 
^(iVp)=|ao(ao+l) + ia,{lO'-^-5(ai-l+97;)+iV,_i + lj- 
The number of digits in 1,. . ., AT is (p+l)(iV+l)-(10^+'-l)/9. See the 
next paper. 
M. d'Ocagne^^ noted that, in writing down the natural numbers 1, . . .,N, 
where N is composed of n digits, the total number of digits written is 
n(iV+l) — In, where 1„ = 1 . . . 1 (to n digits). 
E. Barbier^i" asked what is the W^^%h. digit written if the series of 
natural numbers be written down. 
23Nouv. Ann. Math., (2), 18, 1879, 329; proposed by Laisant, 17, 1878, 480. 
23aAssoc. franQ., 8, 1879, 176-9. 
\ 24Nouv. Corresp. Math., 6, 1880, 519; Mathesis, 1888, 103. 
N s^Les Mondes, 53, 1880, 410-2. 
26NOUV. Corresp. Math., 4, 1878, 387; Nouv. Ann. Math., (3), 2, 1883, 144, 287; 1884, 160. 
»8"Math. Magazine, 1, 1882-4; 69-70; I'intermediaire 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. 
"«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. 
^ojornal de so. math, e ast., 7, 1886, 117-128. 
"/bid., 8, 1887, 101-3; Comptes Rendus Paris, 106, 1888, 190. 
""Comptes Rendus Paris, 105, 1887, 795, 1238. 
