cp's OEIS Frontend

A254648 Numbers n whose square representation in base 10 can be split into three parts whose sum is n.

%I A254648 #22 Aug 27 2017 19:24:35
%S A254648 36,82,91,235,379,414,675,756,792,909,918,964,991,1296,1702,1782,3366,
%T A254648 3646,3682,4132,4906,5149,6832,7543,8416,8767,8856,9208,9325,9586,
%U A254648 9621,9765,9901,9945,9955,9991,12222,12727,17271,22231
%N A254648 Numbers n whose square representation in base 10 can be split into three parts whose sum is n.
%C A254648 Extension of the Kaprekar numbers (A006886) where the number of parts of n^2 is two. It is probably possible to generalize this property with the division of n^2 into m parts.
%C A254648 By convention, the second and third parts may start with the digit 0, but must be positive. For example, 991 is in the sequence because 991^2 = 982081, which can be split into 982, 08 and 1, and 982 + 08 + 1 = 991. But 100 is not; although 100^2 = 10000 and 100 + 0 + 0 = 100, the second and the third part here are not positive. The number 99 is not in the sequence although 99^2 = 9801 and 98 + 0 + 1 = 99.
%C A254648 Property of the sequence:
%C A254648 The sequence is infinite because the numbers of the form 10^n-9 = 91, 991, 991, ... (A170955) are in the sequence: if m = 99...91 with k digits "9", then m^2 = 99...98200...081 with k-1 digits "9" and k-1 digits "0", and 99...982 + 00...8 + 1 = 99...91 = m.
%C A254648 The prime of the sequence are {379, 9901, ...} union {A093177}.
%C A254648 Calculation method: For each class of squares having k-digit numbers, the number of partitions into 3 parts is n(n+1)/2 (A000217). For instance, if the numbers are of the form (abcde) with k = 5, the 6 partitions into 3 subsets are {a,b,{c,d,e}}, {a,{b,c},{d,e}}, {a,{b,c,d},e}, {{a,b},c,{d,e}}, {{a,b},{c,d},e}, {{a,b,c},d,e} and then we compute the corresponding numbers.
%C A254648 Example: 235^2 = 55225 (abcde) = 55225 => {a,b,{c,d,e}} = {5,5,{2,2,5}} => {5,5,225} and 5+5+225 = 235.
%H A254648 Chai Wah Wu, <a href="/A254648/b254648.txt">Table of n, a(n) for n = 1..400</a>
%e A254648 36^2 = 1296 and 1 + 29 + 6 = 36;
%e A254648 235^2 = 55225 and 5 + 5 + 225 = 235;
%e A254648 1782^2 = 3175524 and 3 + 1755 + 24 = 1782;
%e A254648 12727^2 = 161976529 and 1 + 6197 + 6529 = 12727.
%o A254648 (Python)
%o A254648 from itertools import combinations
%o A254648 A254648_list, n, n2 = [], 10, 100
%o A254648 while n < 10**4:
%o A254648     m = str(n2)
%o A254648     for a in combinations(range(1,len(m)),2):
%o A254648         x, y, z = int(m[:a[0]]), int(m[a[0]:a[1]]), int(m[a[1]:])
%o A254648         if y != 0 and z != 0 and x+y+z == n:
%o A254648             A254648_list.append(n)
%o A254648             break
%o A254648     n += 1
%o A254648     n2 += 2*n-1 # _Chai Wah Wu_, Aug 27 2017
%Y A254648 Cf. A000217, A006886, A093177, A170955.
%K A254648 nonn,base
%O A254648 1,1
%A A254648 _Michel Lagneau_, Feb 04 2015
%E A254648 Removed terms 4879 and 5292 by _Chai Wah Wu_, Aug 27 2017