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 A055401 Number of positive cubes needed to sum to n using the greedy algorithm. 14
 0, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 4, 5, 6, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 9, 10, 4, 5, 6, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Define f(n) = n - k^3 where (k+1)^3 > n >= k^3; a(n) = number of steps such that f(f(...f(n)))= 0. Also sum of digits when writing n in base where place values are positive cubes, cf. A000433. [Reinhard Zumkeller, May 08 2011] LINKS Antti Karttunen & Reinhard Zumkeller (terms 1-10000), Table of n, a(n) for n = 0..10000 FORMULA a(0) = 0; for n >= 1, a(n) = a(n-floor(n^(1/3))^3)+1 = a(A055400(n))+1 = a(n-A048762(n))+1. EXAMPLE a(32)=6 because 32=27+1+1+1+1+1 (not 32=8+8+8+8). a(33)=7 because 33=27+1+1+1+1+1+1 (not 33=8+8+8+8+1). MAPLE f:= proc(n, k) local m, j; if n = 0 then return 0 fi; for j from k by -1 while j^3 > n do od: m:= floor(n/j^3); m + procname(n-m*j^3, j-1); end proc: seq(f(n, floor(n^(1/3))), n=0..100); # Robert Israel, Aug 17 2015 MATHEMATICA a[0] = 0; a[n_] := {n} //. {b___, c_ /; !IntegerQ[c^(1/3)], d___} :> {b, f = Floor[c^(1/3)]^3, c - f, d} // Length; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 17 2015 *) PROG (PARI) F=vector(30, n, n^3); /* modify to get other sequences of "greedy representations" */ last_leq(v, F)= { /* Return last element <=v in sorted array F[] */     local(j=1);     while ( F[j]<=v, j+=1 );     return( F[j-1] ); } greedy(n, F)= {     local(v=n, ct=0);     while ( v,  v-=last_leq(v, F); ct+=1; );     return(ct); } vector(min(100, F[#F-1]), n, greedy(n, F)) /* show terms */ /* Joerg Arndt, Apr 08 2011 */ (Haskell) a055401 n = s n \$ reverse \$ takeWhile (<= n) \$ tail a000578_list where   s _ []                 = 0   s m (x:xs) | x > m     = s m xs              | otherwise = m' + s r xs where (m', r) = divMod m x -- Reinhard Zumkeller, May 08 2011 (Scheme, with memoization-macro definec) (definec (A055401 n) (if (zero? n) n (+ 1 (A055401 (A055400 n))))) ;; Antti Karttunen, Aug 16 2015 CROSSREFS Cf. A018888, A055400. Cf. A002376 (least number of positive cubes needed to represent n; differs from this sequence for the first time at n=32, where a(32)=6, while A002376(32)=4). Cf. A053610, A048766, A000578, A000433. Cf. also A261225, A261226, A261227, A261228, A261229. Sequence in context: A053843 A010886 A002376 * A053829 A033928 A194754 Adjacent sequences:  A055398 A055399 A055400 * A055402 A055403 A055404 KEYWORD easy,nonn AUTHOR Henry Bottomley, May 16 2000 EXTENSIONS a(0) = 0 prepended by Antti Karttunen, Aug 16 2015 STATUS approved

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Last modified August 18 21:39 EDT 2017. Contains 290768 sequences.