A358196 Numbers k such that 5^k and 8^k have the same leading digit.

0, 5, 9, 15, 19, 29, 34, 39, 44, 49, 54, 59, 98, 102, 108, 112, 118, 122, 132, 137, 142, 147, 152, 162, 191, 195, 201, 205, 211, 215, 225, 230, 235, 240, 245, 250, 255, 284, 294, 298, 304, 308, 318, 328, 333, 338, 343, 348, 387, 391, 397, 401, 407, 411, 421, 426, 431, 436, 441, 446, 451, 480, 490, 494, 500

Offset: 1

Author

Nicolay Avilov, Nov 02 2022

Keywords

• nonn
• base

Comments

Write lg = log_10, let {x} denote the fractional part of x. Note that {k*lg(5)} = 1 - {k*lg(2)} and that {k*lg(8)} = {k*lg(2)} + 0, 1, or 2, so we have {k > 0 : 5^k and 8^k both start with a} = {k: {k*lg(2)} is in I_a}, where I_a = (1-lg(a+1), 1-lg(a)) intersect (((lg(a))/3, (lg(a+1))/3) U ((lg(a)+1)/3, (lg(a+1)+1)/3) U ((lg(a)+2)/3, (lg(a+1)+2)/3)). Note that I_1 = (1-lg(2), (lg(2)+2)/3), I_3 = ((lg(3)+1)/3, 1-lg(3)), I_5 = (lg(5)/3, lg(6)/3) and that I_a is empty otherwise. As a result, k > 0 is a term if and only if {k*lg(2)} is in (lg(5)/3, lg(6)/3) U ((lg(3)+1)/3, 1-lg(3)) U (1-lg(2), (lg(2)+2)/3). We see that when 5^k and 8^k both start with 1, 3, or 5, 2^k starts with 5, 3, or 1 respectively. - Jianing Song, Dec 26 2022

Examples

			5 is a term because 5^5 = 3125 and 8^5 = 32768;
9 is a term because 5^9 = 1953125 and 8^9 = 134217728.
		

Links

• Nicolay Avilov, Problem 2405. 4 and 5 (in Russian).

Crossrefs

Cf. A000351, A001018, A088935, A111395.

Programs

Mathematica

Select[Range[0, 500], Equal @@ IntegerDigits[{5, 8}^#][[;; , 1]] &] (* Amiram Eldar, Nov 02 2022 *)

PARI

isok(k) = digits(5^k)[1] == digits(8^k)[1]; \\ Michel Marcus, Nov 02 2022

Python

def ok(n): return str(5**n)[0] == str(8**n)[0]
print([k for k in range(501) if ok(k)]) # Michael S. Branicky, Nov 03 2022