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

1,3

Comments

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

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)

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A038206, A102766.

Sequence in context: A068834 A067521 A117686 * A102766 A256349 A202002

Adjacent sequences: A104110 A104111 A104112 * A104114 A104115 A104116

Keyword

nonn,base

Author

Bodo Zinser, Mar 05 2005

Extensions

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

Status

approved