A053610 - OEIS

%I #54 Aug 04 2023 23:16:26

%S 1,2,3,1,2,3,4,2,1,2,3,4,2,3,4,1,2,3,4,2,3,4,5,3,1,2,3,4,2,3,4,5,3,2,

%T 3,1,2,3,4,2,3,4,5,3,2,3,4,5,1,2,3,4,2,3,4,5,3,2,3,4,5,3,4,1,2,3,4,2,

%U 3,4,5,3,2,3,4,5,3,4,5,2,1,2,3,4,2,3,4,5,3,2,3,4,5,3,4,5,2,3,4,1,2,3,4

%N Number of positive squares needed to sum to n using the greedy algorithm.

%C Define f(n) = n - x^2 where (x+1)^2 > n >= x^2. a(n) = number of iterations in f(...f(f(n))...) to reach 0.

%C a(n) = 1 iff n is a perfect square.

%C Also sum of digits when writing n in base where place values are squares, cf. A007961. - _Reinhard Zumkeller_, May 08 2011

%C The sequence could have started with a(0)=0. - _Thomas Ordowski_, Jul 12 2014

%C The sequence is not bounded, see A006892. - _Thomas Ordowski_, Jul 13 2014

%H Reinhard Zumkeller, <a href="/A053610/b053610.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A007953(A007961(n)). - _Henry Bottomley_, Jun 01 2000

%F a(n) = a(n - floor(sqrt(n))^2) + 1 = a(A053186(n)) + 1 [with a(0) = 0]. - _Henry Bottomley_, May 16 2000

%F A053610 = A002828 + A062535. - _M. F. Hasler_, Dec 04 2008

%e 7=4+1+1+1, so 7 requires 4 squares using the greedy algorithm, so a(7)=4.

%p A053610 := proc(n)

%p     local a,x;

%p     a := 0 ;

%p     x := n ;

%p     while x > 0 do

%p         x := x-A048760(x) ;

%p         a := a+1 ;

%p     end do:

%p     a ;

%p end proc: # _R. J. Mathar_, May 13 2016

%t f[n_] := (n - Floor[Sqrt[n]]^2); g[n_] := (m = n; c = 1; While[a = f[m]; a != 0, c++; m = a]; c); Table[ g[n], {n, 1, 105}]

%o (PARI) A053610(n,c=1)=while(n-=sqrtint(n)^2,c++);c \\ _M. F. Hasler_, Dec 04 2008

%o (Haskell)

%o a053610 n = s n $ reverse $ takeWhile (<= n) $ tail a000290_list where

%o   s _ []                 = 0

%o   s m (x:xs) | x > m     = s m xs

%o              | otherwise = m' + s r xs where (m',r) = divMod m x

%o -- _Reinhard Zumkeller_, May 08 2011

%o (Python)

%o from math import isqrt

%o def A053610(n):

%o     c = 0

%o     while n:

%o         n -= isqrt(n)**2

%o         c += 1

%o     return c # _Chai Wah Wu_, Aug 01 2023

%Y Cf. A006892 (positions of records), A055401, A007961.

%Y Cf. A000196, A000290, A057945 (summing triangular numbers).

%K nonn

%O 1,2

%A _Jud McCranie_, Mar 19 2000