A055401 -id:A055401 - cp's OEIS frontend

A261225 n minus the number of positive cubes needed to sum to n using the greedy algorithm: a(n) = n - A055401(n).

0, 0, 0, 0, 0, 0, 0, 0, 7, 7, 7, 7, 7, 7, 7, 7, 14, 14, 14, 14, 14, 14, 14, 14, 21, 21, 21, 26, 26, 26, 26, 26, 26, 26, 26, 33, 33, 33, 33, 33, 33, 33, 33, 40, 40, 40, 40, 40, 40, 40, 40, 47, 47, 47, 52, 52, 52, 52, 52, 52, 52, 52, 59, 59, 63, 63, 63, 63, 63, 63, 63, 63, 70, 70, 70, 70, 70, 70, 70, 70, 77, 77, 77, 77, 77, 77, 77, 77, 84, 84, 84, 89

Offset: 0

Antti Karttunen, Aug 16 2015

nonn

			a(8) = 7, because when the greedy algorithm partitions 8 into cubes, it first finds 8 (= 2*2*2), thus A055401(8) = 1, and 8-1 = 7.

Antti Karttunen, Table of n, a(n) for n = 0..12167

Cf. A048762, A055401, A261226, A261227, A261228, A261229.

Cf. also A260740.

a(n) = n - A055401(n).
As a recurrence:
a(0) = 0; for n >= 1, a(n) = -1 + A048762(n) + a(n-A048762(n)). [Where A048762(n) gives the largest cube <= n.]

A261228 a(n) = number of steps to reach 0 when starting from k = (n^3)-1 and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

Original entry on oeis.org

0, 1, 4, 10, 19, 33, 52, 77, 108, 146, 190, 244, 306, 377, 458, 549, 652, 767, 896, 1038, 1195, 1367, 1554, 1757, 1978, 2216, 2472, 2746, 3040, 3353, 3688, 4045, 4423, 4823, 5247, 5696, 6169, 6668, 7193, 7745, 8324, 8933, 9570, 10236, 10934, 11663, 12423, 13215, 14042, 14902, 15797, 16726, 17693, 18695, 19734, 20811, 21928, 23083, 24278, 25513

Offset: 1

Antti Karttunen, Aug 16 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..512

One less than A261227.
First differences: A261229.
Cf. A000578, A055401, A261225, A261226.
Cf. also A261223.

a(1) = 0; for n > 1, a(n) = A261229(n-1) + a(n-1).

a(n) = A261226((n^3)-1).

A261229 a(n) = number of steps to reach (n^3)-1 when starting from k = ((n+1)^3)-1 and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

Original entry on oeis.org

1, 3, 6, 9, 14, 19, 25, 31, 38, 44, 54, 62, 71, 81, 91, 103, 115, 129, 142, 157, 172, 187, 203, 221, 238, 256, 274, 294, 313, 335, 357, 378, 400, 424, 449, 473, 499, 525, 552, 579, 609, 637, 666, 698, 729, 760, 792, 827, 860, 895, 929, 967, 1002, 1039, 1077, 1117, 1155, 1195, 1235, 1278, 1318, 1361, 1404, 1448, 1492, 1538, 1583, 1631, 1677, 1725

Offset: 1

Antti Karttunen, Aug 16 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..512

First differences of both A261227 and A261228.
Cf. A000578, A055401, A261225, A261226.
Cf. also A261224.

a(n) = A261226(((n+1)^3)-1) - A261226((n^3)-1). [The definition.]
Equally, for all n >= 1:
a(n) = A261226((n+1)^3) - A261226(n^3).
a(n) = A261227(n+1) - A261227(n).
a(n) = A261228(n+1) - A261228(n).

A276613 The infinite trunk of greedy cubes beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of cubes that sum to a(n) with greedy algorithm (A055401).

Original entry on oeis.org

0, 7, 14, 21, 26, 33, 40, 47, 52, 59, 63, 70, 77, 84, 89, 96, 103, 110, 115, 124, 131, 138, 145, 150, 157, 164, 171, 176, 183, 187, 194, 201, 208, 215, 222, 229, 236, 241, 248, 255, 262, 267, 274, 278, 285, 292, 299, 304, 311, 318, 330, 339, 342, 349, 356, 363, 368, 375, 382, 389, 394, 401, 405, 412, 419, 426, 431, 438, 445

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10236

Cf. A055401, A261225, A276612, A276614, A276615 (first differences).

Cf. also A179016, A259934, A276573, A276583, A276623 for similar constructions.

(define (A276613 n) (A276614 (A276612 n)))

a(n) = A276614(A276612(n)).

A261227 a(n) = number of steps to reach 0 when starting from k = n^3 and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

Original entry on oeis.org

0, 1, 2, 5, 11, 20, 34, 53, 78, 109, 147, 191, 245, 307, 378, 459, 550, 653, 768, 897, 1039, 1196, 1368, 1555, 1758, 1979, 2217, 2473, 2747, 3041, 3354, 3689, 4046, 4424, 4824, 5248, 5697, 6170, 6669, 7194, 7746, 8325, 8934, 9571, 10237, 10935, 11664, 12424, 13216, 14043, 14903, 15798, 16727, 17694, 18696, 19735, 20812, 21929, 23084, 24279, 25514

Offset: 0

Antti Karttunen, Aug 16 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..512

Essentially one more than A261228.
First differences: A261229.
Cf. A000578, A055401, A261225, A261226.
Cf. also A261222.

a(0) = 0, a(1) = 1; for n >= 2, a(n) = A261229(n-1) + a(n-1).

a(n) = A261226(n^3).

A261226 a(n) = number of steps to reach 0 when starting from k = n and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16

Offset: 0

Antti Karttunen, Aug 16 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..12167

Cf. A055401, A261225, A261227, A261228, A261229.
Cf. also A261221.
After a(0) differs from A003108 for the first time at n=32, where a(32)=5, while A003108(32)=6.

(definec (A261226 n) (if (zero? n) n (+ 1 (A261226 (A261225 n)))))

a(0) = 0; for n >= 1, a(n) = 1 + a(A261225(n)).

A053610 Number of positive squares needed to sum to n using the greedy algorithm.

Original entry on oeis.org

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, 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, 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

Offset: 1

Jud McCranie, Mar 19 2000

nonn

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.
a(n) = 1 iff n is a perfect square.
Also sum of digits when writing n in base where place values are squares, cf. A007961. - Reinhard Zumkeller, May 08 2011
The sequence could have started with a(0)=0. - Thomas Ordowski, Jul 12 2014
The sequence is not bounded, see A006892. - Thomas Ordowski, Jul 13 2014

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

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

a053610 n = s n $ reverse $ takeWhile (<= n) $ tail a000290_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

A053610 := proc(n)
    local a,x;
    a := 0 ;
    x := n ;
    while x > 0 do
        x := x-A048760(x) ;
        a := a+1 ;
    end do:
    a ;
end proc: # R. J. Mathar, May 13 2016

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}]

A053610(n,c=1)=while(n-=sqrtint(n)^2,c++);c \\ M. F. Hasler, Dec 04 2008

from math import isqrt
def A053610(n):
    c = 0
    while n:
        n -= isqrt(n)**2
        c += 1
    return c # Chai Wah Wu, Aug 01 2023

a(n) = A007953(A007961(n)). - Henry Bottomley, Jun 01 2000
a(n) = a(n - floor(sqrt(n))^2) + 1 = a(A053186(n)) + 1 [with a(0) = 0]. - Henry Bottomley, May 16 2000
A053610 = A002828 + A062535. - M. F. Hasler, Dec 04 2008

A002376 Least number of positive cubes needed to sum to n.

Original entry on oeis.org

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, 4, 5, 6, 2, 3, 4, 5, 6, 5, 6, 7, 3, 4, 5, 6, 7, 6, 7, 8, 4, 5, 6, 2, 3, 4, 5, 6, 5, 6, 7, 3, 4, 1, 2, 3, 4, 5, 6, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 3, 3, 4, 5, 6, 7, 6, 7, 4, 4, 5, 2, 3, 4, 5, 6, 5, 5, 6, 3, 4, 5, 6, 7, 6, 6

Offset: 1

N. J. A. Sloane

No terms are greater than 9, see A002804. - Charles R Greathouse IV, Aug 01 2013

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 81.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. R. Zornow, De compositione numerorum e cubis integris positivus, J. Reine Angew. Math., 14 (1835), 276-280.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Cubic Number

Cf. A000578, A003325 (numbers requiring 2 cubes), A047702 (numbers requiring 3 cubes), A047703 (numbers requiring 4 cubes), A047704 (numbers requiring 5 cubes), A046040 (numbers requiring 6 cubes), A018890 (numbers requiring 7 cubes), A018888 (numbers requiring 8 or 9 cubes), A055401 (cubes needed by greedy algorithm).

f:= proc(n) option remember;
  min(seq(procname(n - i^3)+1, i=1..floor(n^(1/3))))
end proc:
f(0):= 0:
map(f, [$1..100]); # Robert Israel, Jun 30 2017

CubesCnt[n_] := Module[{k = 1}, While[Length[PowersRepresentations[n, k, 3]] == 0, k++]; k]; Array[CubesCnt, 100] (* T. D. Noe, Apr 01 2011 *)

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A002376(n):
    if n == 1: return 1
    for k in count(1):
        try:
            next(power_representation(n,3,k))
        except:
            continue
        return k # Chai Wah Wu, Jun 25 2024

The g.f. conjectured by Simon Plouffe in his 1992 dissertation,

-(-1-z-z^2-z^3-z^4-z^5-z^6+6*z^7)/(z+1)/(z^2+1)/(z^4+1)/(z-1)^2, is incorrect: the first wrong coefficient is that of z^26. - Robert Israel, Jun 30 2017

More terms from Arlin Anderson (starship1(AT)gmail.com)

A055500 a(0)=1, a(1)=1, a(n) = largest prime <= a(n-1) + a(n-2).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 17, 23, 37, 59, 89, 139, 227, 359, 577, 929, 1499, 2423, 3919, 6337, 10253, 16573, 26821, 43391, 70207, 113591, 183797, 297377, 481171, 778541, 1259701, 2038217, 3297913, 5336129, 8633983, 13970093, 22604069, 36574151, 59178199, 95752333

Offset: 0

N. J. A. Sloane, Jul 08 2000

Or might be called Ishikawa primes, as he proved that prime(n+2) < prime(n) + prime(n+1) for n > 1. This improves on Bertrand's Postulate (Chebyshev's theorem), which says prime(n+2) < prime(n+1) + prime(n+1). - Jonathan Sondow, Sep 21 2013

			a(8) = 23 because 23 is largest prime <= a(7) + a(6) = 17 + 11 = 28.

Michael De Vlieger, Table of n, a(n) for n = 0..1000 (first 100 terms from Zak Seidov)
Heihachiro Ishikawa, Über die Verteilung der Primzahlen, Sci. Rep. Tokyo Bunrika Daigaku, Sect. A 2 (1934), 27-40.

Cf. A007917, A055498, A055499, A055400, A055401, A055502, A065435.

a055500 n = a055500_list !! n
a055500_list = 1 : 1 : map a007917
               (zipWith (+) a055500_list $ tail a055500_list)
-- Reinhard Zumkeller, May 01 2013

PrevPrim[n_] := Block[ {k = n}, While[ !PrimeQ[k], k-- ]; Return[k]]; a[1] = a[2] = 1; a[n_] := a[n] = PrevPrim[ a[n - 1] + a[n - 2]]; Table[ a[n], {n, 1, 42} ]
(* Or, if version >= 6 : *)a[0] = a[1] = 1; a[n_] := a[n] = NextPrime[ a[n-1] + a[n-2] + 1, -1]; Table[a[n], {n, 0, 100}](* Jean-François Alcover, Jan 12 2012 *)
nxt[{a_,b_}]:={b,NextPrime[a+b+1,-1]}; Transpose[NestList[nxt,{1,1},40]] [[1]] (* Harvey P. Dale, Jul 15 2013 *)

from sympy import prevprime; L = [1, 1]
for _ in range(36): L.append(prevprime(L[-2] + L[-1] + 1))
print(*L, sep = ", ")  # Ya-Ping Lu, May 05 2023

a(n) is asymptotic to C*phi^n where phi = (1+sqrt(5))/2 and C = 0.41845009129953131631777132510164822489... - Benoit Cloitre, Apr 21 2003
a(n) = A007917(a(n-1) + a(n-2)) for n > 1. - Reinhard Zumkeller, May 01 2013
a(n) >= prime(n-1) for n > 1, by Ishikawa's theorem. - Jonathan Sondow, Sep 21 2013

A055400 Cube excess: difference between n and largest cube <= n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset: 0

Henry Bottomley, May 16 2000

			a(12) = 4 because 2^3 <= 12 < 3^3 and 12 - 2^3 = 4.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A048762, A053186, A055401.

A055400 := proc(n)
        n-A048762(n) ;
end proc:
seq(A055400(n),n=0..80) ; # R. J. Mathar, Nov 06 2011

Array[#-Floor[Power[#, (3)^-1]]^3&,90,0] (* Harvey P. Dale, Jan 18 2012 *)

a(n)=n-sqrtnint(n,3)^3 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = n - A048762(n) = n - floor(n^(1/3))^3.

a(n) < 3*n^(2/3) for n > 0. - Charles R Greathouse IV, Sep 02 2015

cp's OEIS Frontend

A261225 n minus the number of positive cubes needed to sum to n using the greedy algorithm: a(n) = n - A055401(n).

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A261228 a(n) = number of steps to reach 0 when starting from k = (n^3)-1 and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

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A261229 a(n) = number of steps to reach (n^3)-1 when starting from k = ((n+1)^3)-1 and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

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A276613 The infinite trunk of greedy cubes beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of cubes that sum to a(n) with greedy algorithm (A055401).

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A261227 a(n) = number of steps to reach 0 when starting from k = n^3 and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

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A261226 a(n) = number of steps to reach 0 when starting from k = n and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

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A053610 Number of positive squares needed to sum to n using the greedy algorithm.

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Maple

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A002376 Least number of positive cubes needed to sum to n.

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A055500 a(0)=1, a(1)=1, a(n) = largest prime <= a(n-1) + a(n-2).

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