A379980 Numbers that are divisible by the square of the sum of the squares of their digits.

1, 10, 100, 1000, 1100, 1200, 1300, 2000, 2023, 2100, 2400, 3100, 4332, 5000, 10000, 10100, 10200, 10300, 11000, 12000, 13000, 20000, 20100, 20230, 20400, 21000, 24000, 30100, 30324, 31000, 31311, 42000, 43011, 43320, 50000, 52022, 52215, 55000, 71824, 100000

Offset: 1

Author

Amiram Eldar, Jan 07 2025

Keywords

• nonn
• base
• easy

Comments

Called "Second-order Harshad numbers" by Pal and Gopalan (2023).

If k is a term, then 10*k is also a term.

Examples

			10 is a term since 10 is divisible by (1^2 + 0^2)^2 = 1.
1100 is a term since 1100 is divisible by (1^2 + 1^2 + 0^2 + 0^2)^2 = 4.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Pradip Kumar Pal and Kaushik Gopalan, Second Order Harshad Number, International Journal of Mathematical Education, Vol. 13, No. 1 (2023), pp. 25-26.

Crossrefs

Cf. A003132, A005349, A072081, A180490 (binary analog).

Subsequence of A034087.

Subsequences: A379981, A379982.

Programs

Mathematica

Select[Range[10^5], Divisible[#, (Plus @@ (IntegerDigits[#]^2))^2] &]

PARI

isok(k) = !(k % vecsum(apply(x -> x^2, digits(k)))^2);

Python

def ok(n): return n and n%sum(di**2 for di in map(int, str(n)))**2 == 0
print([k for k in range(100001) if ok(k)]) # Michael S. Branicky, Jan 10 2025