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A153745
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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.
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10
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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
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OFFSET
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1,2
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COMMENTS
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This sequence is a subsequence of A061910.
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LINKS
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FORMULA
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EXAMPLE
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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.
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PROG
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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
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:
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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