A104113 Numbers which when chopped into one, two or more parts, added and squared result in the same number.

0, 1, 81, 100, 1296, 2025, 3025, 6724, 8281, 9801, 10000, 55225, 88209, 136161, 136900, 143641, 171396, 431649, 455625, 494209, 571536, 627264, 826281, 842724, 893025, 929296, 980100, 982081, 998001, 1000000, 1679616, 2896804, 3175524, 4941729, 7441984

Offset: 1

Author

Bodo Zinser, Mar 05 2005

Keywords

• nonn
• base

Comments

Every term is congruent to 0 or 1 modulo 9. - Andrea Tarantini, Sep 27 2021

Examples

			1296 is a term since (1+29+6)^2 = 36^2 = 1296.
		

Links

• John Drake, Table of n, a(n) for n = 1..408 (terms 1..80 from Mehrad Mahmoudian, terms 81..225 from Giovanni Resta)
• Mehrad Mahmoudian, R code to produce the sequence
• Mehrad Mahmoudian, Decompositions for a(1)-a(80)
• Project Euler, Problem 719: Number Splitting (2020)

Crossrefs

Cf. A038206, A102766.

Programs

Mathematica

Join[{0},Select[Select[Range@3000^2,Mod[#,9]<2&],(n=#;MemberQ[(Total/@(FromDigits/@#&/@Union[DeleteCases[SplitBy[#,#==-1&],{-1}]&/@(Insert[IntegerDigits@n,-1,#]&/@(List/@#&/@Rest@Subsets[Range@IntegerLength@n]))]))^2,#])&]] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

Python

def expr(t, d): # can you express target t with digits d, only adding +'s
    if t < 0: return False
    if t == int(d): return True
    return any(expr(t-int(d[:i]), d[i:]) for i in range(1, len(d)))
def aupto(limit):
    alst, k, k2 = [], 0, 0
    while k2 <= limit:
        if expr(k, str(k2)):
            alst.append(k2)
        k, k2 = k+1, k2 + 2*k + 1
    return alst
print(aupto(7500000)) # Michael S. Branicky, Sep 27 2021

Formula

a(n) = A038206(n)^2. - Andrea Tarantini, Sep 27 2021

Extensions

a(30) and beyond from Mehrad Mahmoudian, Dec 16 2019