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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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%I #37 May 09 2021 07:52:33

%S 1,2,3,39,60,86,90,321,347,401,3387,3414,3578,3900,4767,6000,6549,

%T 6552,6744,6780,6783,7387,7862,7889,8367,8598,8600,8773,8898,9000,

%U 9220,9884,9885,10000,10001,10002,10003,10004,10005,10010,10011,10012,10013,10020

%N 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.

%C This sequence is a subsequence of A061910.

%H Chai Wah Wu, <a href="/A153745/b153745.txt">Table of n, a(n) for n = 1..3749</a>

%F a(n) = sqrt(A258660(n)). - _Doug Bell_, Jun 15 2015

%e 39^2 = 1521; 1+5+2+1 = 9 = 3^2 and 15+21 = 36 = 6^2.

%e 321^2 = 103041; 1+0+3+0+4+1 = 9 = 3^2; 10+30+41 = 81 = 9^2; and 103+041 = 144 = 12^2.

%o (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

%o (Python)

%o from sympy import divisors

%o from gmpy2 import is_prime, isqrt_rem, isqrt, is_square

%o A153745_list = []

%o for l in range(1,20):

%o ....if not is_prime(l):

%o ........fs = divisors(l)

%o ........a, b = isqrt_rem(10**(l-1))

%o ........if b > 0:

%o ............a += 1

%o ........for n in range(a,isqrt(10**l-1)+1):

%o ............ns = str(n**2)

%o ............for g in fs:

%o ................y = 0

%o ................for h in range(0,l,g):

%o ....................y += int(ns[h:h+g])

%o ................if not is_square(y):

%o ....................break

%o ............else:

%o ................A153745_list.append(n) # _Chai Wah Wu_, Jun 08 2015

%Y Cf. A004159, A061910, A258660.

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

%K nonn,base

%O 1,2

%A _Doug Bell_, Dec 31 2008

%E Data corrected by _Doug Bell_, Jan 19 2009

%E Name corrected by _Doug Bell_, Jun 06 2015