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
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
Programs
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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
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