A104113 - OEIS

%I #60 Jul 12 2023 11:12:44

%S 0,1,81,100,1296,2025,3025,6724,8281,9801,10000,55225,88209,136161,

%T 136900,143641,171396,431649,455625,494209,571536,627264,826281,

%U 842724,893025,929296,980100,982081,998001,1000000,1679616,2896804,3175524,4941729,7441984

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

%C Every term is congruent to 0 or 1 modulo 9. - _Andrea Tarantini_, Sep 27 2021

%H John Drake, <a href="/A104113/b104113.txt">Table of n, a(n) for n = 1..408</a> (terms 1..80 from Mehrad Mahmoudian, terms 81..225 from Giovanni Resta)

%H Mehrad Mahmoudian, <a href="https://gist.github.com/mmahmoudian/32dbd85442e74e93b733d4fe48beda1e">R code to produce the sequence</a>

%H Mehrad Mahmoudian, <a href="/A104113/a104113_1.txt">Decompositions for a(1)-a(80)</a>

%H Project Euler, <a href="https://projecteuler.net/problem=719">Problem 719: Number Splitting</a> (2020)

%F a(n) = A038206(n)^2. - _Andrea Tarantini_, Sep 27 2021

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

%t 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 *)

%o (Python)

%o def expr(t, d): # can you express target t with digits d, only adding +'s

%o     if t < 0: return False

%o     if t == int(d): return True

%o     return any(expr(t-int(d[:i]), d[i:]) for i in range(1, len(d)))

%o def aupto(limit):

%o     alst, k, k2 = [], 0, 0

%o     while k2 <= limit:

%o         if expr(k, str(k2)):

%o             alst.append(k2)

%o         k, k2 = k+1, k2 + 2*k + 1

%o     return alst

%o print(aupto(7500000)) # _Michael S. Branicky_, Sep 27 2021

%Y Cf. A038206, A102766.

%K nonn,base

%O 1,3

%A _Bodo Zinser_, Mar 05 2005

%E a(30) and beyond from _Mehrad Mahmoudian_, Dec 16 2019