A258660 Numbers n such that the number of digits d in n is not prime and for each factor f of d the sum of the d/f digit groupings of size f is a square.

1, 4, 9, 1521, 3600, 7396, 8100, 103041, 120409, 160801, 11471769, 11655396, 12802084, 15210000, 22724289, 36000000, 42889401, 42928704, 45481536, 45968400, 46009089, 54567769, 61811044, 62236321, 70006689, 73925604, 73960000, 76965529, 79174404, 81000000, 85008400, 97693456, 97713225, 100000000

Offset: 1

Author

Doug Bell, Jun 06 2015

Keywords

• base
• nonn

Comments

If a(n) has m = p^k digits, then a(n)*10^((p-1)*m) is also a member of the sequence. For instance, 1521*10^(2^k-4) is in the sequence for all integers k >=2. # Chai Wah Wu, Jun 08 2015

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..3730

Crossrefs

Cf. A153745.

Programs

Python

from sympy import divisors
from gmpy2 import is_prime, isqrt, isqrt_rem, is_square
A258660_list = []
for l in range(1,17):
    if not is_prime(l):
        fs = divisors(l)
        a, b = isqrt_rem(10**(l-1))
        if b > 0:
            a += 1
        for n in range(a,isqrt(10**l-1)+1):
            n2 = n**2
            ns = str(n2)
            for g in fs:
                y = 0
                for h in range(0,l,g):
                    y += int(ns[h:h+g])
                if not is_square(y):
                    break
            else:
                A258660_list.append(n2) # Chai Wah Wu, Jun 08 2015

Formula

a(n) = A153745(n)^2.

Extensions

Corrected a(13)-a(14) by Chai Wah Wu, Jun 08 2015