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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.
2

%I #12 Jun 09 2015 21:07:58

%S 1,4,9,1521,3600,7396,8100,103041,120409,160801,11471769,11655396,

%T 12802084,15210000,22724289,36000000,42889401,42928704,45481536,

%U 45968400,46009089,54567769,61811044,62236321,70006689,73925604,73960000,76965529,79174404,81000000,85008400,97693456,97713225,100000000

%N 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.

%C 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

%H Chai Wah Wu, <a href="/A258660/b258660.txt">Table of n, a(n) for n = 1..3730</a>

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

%o (Python)

%o from sympy import divisors

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

%o A258660_list = []

%o for l in range(1,17):

%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 ............n2 = n**2

%o ............ns = str(n2)

%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 ................A258660_list.append(n2) # _Chai Wah Wu_, Jun 08 2015

%Y Cf. A153745.

%K base,nonn

%O 1,2

%A _Doug Bell_, Jun 06 2015

%E Corrected a(13)-a(14) by _Chai Wah Wu_, Jun 08 2015