A348300 a(n) is the largest number that is the digit sum of the square of an n-digit number.

13, 31, 46, 63, 81, 97, 112, 130, 148, 162, 180, 193, 211, 229, 244, 262, 277, 297, 310, 331, 343, 360, 378, 396

Offset: 1

Author

Bernardo Recamán and Freddy Barrera, Oct 10 2021

Keywords

• nonn
• base
• more

Comments

18*n-a(n) appears to be nondecreasing. - Chai Wah Wu, Nov 18 2021

According to new data 18*n-a(n) sometimes decreases. - David A. Corneth, Feb 21 2024

a(n) is the digit sum of the square of the last n-digit integer in A067179. - Zhao Hui Du, Mar 04 2024

a(n) appears to be approximately equal to 16.5*n. - Zhining Yang, Mar 12 2024

a(n) modulo 9 is either 0, 1, 4 or 7. - Chai Wah Wu, Apr 04 2024

Examples

			a(3) = 46 because 46 is the largest digital sum encountered among the squares (that of 937) of all 3-digit numbers. Such maximal digital sum can be achieved by more than one square (squares of 836 and 883 also have digital sum 46). Largest of these is A348303.
		

Crossrefs

Cf. A004159, A348303, A370522.

Cf. A371728.

Programs

Mathematica

Array[Max@ Map[Total@ IntegerDigits[#^2] &, Range[10^(# - 1), 10^# - 1]] &, 8] (* Michael De Vlieger, Oct 12 2021 *)

Python

def A348300(n): return max(sum(int(d) for d in str(m**2)) for m in range(10**(n-1),10**n)) # Chai Wah Wu, Jun 26 2024

Sage

def A348300(n):
    return max(sum((k^2).digits()) for k in (10^(n-1)..10^n-1))

Formula

a(n) = Max_{k=10^(n-1)..10^n-1} A004159(k).

Extensions

a(11) from Chai Wah Wu, Nov 18 2021

a(12)-a(13) from Martin Ehrenstein, Nov 20 2021

a(14)-a(24) from Zhao Hui Du, Feb 23 2024

Name edited by Jon E. Schoenfield, Mar 10 2024