A385827 a(n) is the unique k such that 2^k is closest to 5^n.

0, 2, 5, 7, 9, 12, 14, 16, 18, 21, 23, 25, 28, 30, 32, 35, 37, 39, 42, 44, 46, 49, 51, 53, 56, 58, 60, 63, 65, 67, 70, 72, 74, 77, 79, 81, 84, 86, 88, 90, 93, 95, 97, 100, 102, 104, 107, 109, 111, 114, 116, 118, 121, 123, 125, 128, 130, 132, 135, 137, 139, 142, 144, 146, 149

Offset: 0

Author

Richard S. Chang, Jul 09 2025

Keywords

• easy
• nonn

Comments

a(n) is the unique k such that 2^k/5^n is closest to 1.

For all n, 2/3 < 2^a(n)/5^n < 4/3.

For n > 0, a(n) = A061785(n) if log_2(2*5^n/3) < floor(n*log_2(5)), otherwise a(n) = A061785(n) + 1.

Examples

			a(3) = 7 because 2^7 = 128 is the closest power of 2 to 5^3 = 125.
		

Links

• A. Gorman-Huang and E. Marks, Approximating Powers of 2 Using Powers of 3 and Break Point Rounding, Girls' Angle Bulletin, Vol. 18, No. 5 (2025), 8-10.

Crossrefs

Cf. A020857, A061785.

Programs

Mathematica

a[n_]:=-Floor[-n*Log2[5] + Log2[3/2]]; Array[a,65,0] (* Stefano Spezia, Jul 17 2025 *)

Python

def A385827(n):
    if n == 0: return 0
    k = 5**n
    m = k.bit_length()-2
    return m+1+(k>3<Chai Wah Wu, Jul 22 2025

Formula

a(n) = -floor(-n*log_2(5) + log_2(3/2)).