cp's OEIS Frontend

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.

%I A153745 #39 May 22 2025 10:21:35
%S A153745 1,2,3,39,60,86,90,321,347,401,3387,3414,3578,3900,4767,6000,6549,
%T A153745 6552,6744,6780,6783,7387,7862,7889,8367,8598,8600,8773,8898,9000,
%U A153745 9220,9884,9885,10000,10001,10002,10003,10004,10005,10010,10011,10012,10013,10020
%N 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.
%C A153745 This sequence is a subsequence of A061910.
%H A153745 Chai Wah Wu, <a href="/A153745/b153745.txt">Table of n, a(n) for n = 1..3749</a>
%F A153745 a(n) = sqrt(A258660(n)). - _Doug Bell_, Jun 15 2015
%e A153745 39^2 = 1521; 1+5+2+1 = 9 = 3^2 and 15+21 = 36 = 6^2.
%e A153745 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 A153745 (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 A153745 (Python)
%o A153745 from sympy import divisors
%o A153745 from gmpy2 import is_prime, isqrt_rem, isqrt, is_square
%o A153745 A153745_list = []
%o A153745 for l in range(1,20):
%o A153745     if not is_prime(l):
%o A153745         fs = divisors(l)
%o A153745         a, b = isqrt_rem(10**(l-1))
%o A153745         if b > 0:
%o A153745             a += 1
%o A153745         for n in range(a,isqrt(10**l-1)+1):
%o A153745             ns = str(n**2)
%o A153745             for g in fs:
%o A153745                 y = 0
%o A153745                 for h in range(0,l,g):
%o A153745                     y += int(ns[h:h+g])
%o A153745                 if not is_square(y):
%o A153745                     break
%o A153745             else:
%o A153745                 A153745_list.append(n) # _Chai Wah Wu_, Jun 08 2015
%Y A153745 Cf. A004159, A061910, A258660.
%Y A153745 Subsequences: A153746, A153747, A153748, A153749, A153750, A153751, A153752, A153753.
%K A153745 nonn,base
%O A153745 1,2
%A A153745 _Doug Bell_, Dec 31 2008
%E A153745 Data corrected by _Doug Bell_, Jan 19 2009
%E A153745 Name corrected by _Doug Bell_, Jun 06 2015