A153745 Numbers k such that the number of digits d in k^2 is not prime and for each factor f of d the sum of the d/f digit groupings in k^2 of size f is a square.

1, 2, 3, 39, 60, 86, 90, 321, 347, 401, 3387, 3414, 3578, 3900, 4767, 6000, 6549, 6552, 6744, 6780, 6783, 7387, 7862, 7889, 8367, 8598, 8600, 8773, 8898, 9000, 9220, 9884, 9885, 10000, 10001, 10002, 10003, 10004, 10005, 10010, 10011, 10012, 10013, 10020

Offset: 1

Author

Doug Bell, Dec 31 2008

Keywords

• nonn
• base

Comments

This sequence is a subsequence of A061910.

Examples

			39^2 = 1521; 1+5+2+1 = 9 = 3^2 and 15+21 = 36 = 6^2.
321^2 = 103041; 1+0+3+0+4+1 = 9 = 3^2; 10+30+41 = 81 = 9^2; and 103+041 = 144 = 12^2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..3749

Crossrefs

Cf. A004159, A061910, A258660.

Subsequences: A153746, A153747, A153748, A153749, A153750, A153751, A153752, A153753.

Programs

PARI

isok(n) = {my(d = digits(n^2)); if (! isprime(#d), my(dd = divisors(#d)); for (k=1, #dd, my(tg = 10^dd[k]); my(s = 0); my(m = n^2); for (i=1, #d/dd[k], s += m % tg; m = m\tg;); if (! issquare(s), return(0));); return (1););} \\ Michel Marcus, Jun 06 2015

Python

from sympy import divisors
from gmpy2 import is_prime, isqrt_rem, isqrt, is_square
A153745_list = []
for l in range(1,20):
    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):
            ns = str(n**2)
            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:
                A153745_list.append(n) # Chai Wah Wu, Jun 08 2015

Formula

a(n) = sqrt(A258660(n)). - Doug Bell, Jun 15 2015

Extensions

Data corrected by Doug Bell, Jan 19 2009

Name corrected by Doug Bell, Jun 06 2015