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

36, 82, 91, 235, 379, 414, 675, 756, 792, 909, 918, 964, 991, 1296, 1702, 1782, 3366, 3646, 3682, 4132, 4906, 5149, 6832, 7543, 8416, 8767, 8856, 9208, 9325, 9586, 9621, 9765, 9901, 9945, 9955, 9991, 12222, 12727, 17271, 22231

Offset

1,1

Comments

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.

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.

Property of the sequence:

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.

The prime of the sequence are {379, 9901, ...} union {A093177}.

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.

Example: 235^2 = 55225 (abcde) = 55225 => {a,b,{c,d,e}} = {5,5,{2,2,5}} => {5,5,225} and 5+5+225 = 235.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..400

Example

36^2 = 1296 and 1 + 29 + 6 = 36;
235^2 = 55225 and 5 + 5 + 225 = 235;
1782^2 = 3175524 and 3 + 1755 + 24 = 1782;
12727^2 = 161976529 and 1 + 6197 + 6529 = 12727.

Prog

(Python)
from itertools import combinations
A254648_list, n, n2 = [], 10, 100
while n < 10**4:
    m = str(n2)
    for a in combinations(range(1, len(m)), 2):
        x, y, z = int(m[:a[0]]), int(m[a[0]:a[1]]), int(m[a[1]:])
        if y != 0 and z != 0 and x+y+z == n:
            A254648_list.append(n)
            break
    n += 1
    n2 += 2*n-1 # Chai Wah Wu, Aug 27 2017

Crossrefs

Cf. A000217, A006886, A093177, A170955.

Sequence in context: A259240 A341555 A084006 * A202330 A226264 A243804

Adjacent sequences: A254645 A254646 A254647 * A254649 A254650 A254651

Keyword

nonn,base

Author

Michel Lagneau, Feb 04 2015

Extensions

Removed terms 4879 and 5292 by Chai Wah Wu, Aug 27 2017

Status

approved