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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. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence is a subsequence of A061910.

LINKS

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

FORMULA

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

EXAMPLE

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.

PROG

(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

CROSSREFS

Cf. A004159, A061910, A258660.

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

Sequence in context: A060813 A039820 A331204 * A076724 A080393 A111683

Adjacent sequences: A153742 A153743 A153744 * A153746 A153747 A153748

KEYWORD

nonn,base

AUTHOR

Doug Bell, Dec 31 2008

EXTENSIONS

Data corrected by Doug Bell, Jan 19 2009

Name corrected by Doug Bell, Jun 06 2015

STATUS

approved

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Last modified November 28 09:43 EST 2022. Contains 358407 sequences. (Running on oeis4.)