A054966 -id:A054966 - cp's OEIS frontend

A061096 Let k = n-th number that is a possible digit-sum for a cube (A054966); sequence gives smallest cube with digit-sum k.

1, 8, 27, 64, 2744, 729, 2197, 17576, 19683, 6859, 148877, 287496, 438976, 778688, 2299968, 3869893, 43986977, 75686967, 174676879, 596947688, 796597983, 1693669888, 9649992689, 56888939736, 7598896696, 78898389569, 197747699976, 677298787768, 1778597976896

Offset: 0

Amarnath Murthy, Apr 19 2001

			a(5) = 2744, sum of digits = 17, the fifth term of A054966 (1,8,9,10,17,18...)

Amarnath Murthy, Fabricating a perfect cube with a given valid digit sum (to be published)

Cf. A054966, A067177.

a(n) = A067177(n)^3. - R. J. Mathar, Aug 23 2018

More terms from Sascha Kurz, Jan 28 2003

A135043 Duplicate of A054966.

Original entry on oeis.org

0, 1, 8, 9, 10, 17, 18, 19, 26, 27, 28, 35, 36, 37, 44, 45, 46, 53, 54, 55, 62, 63, 64, 71, 72, 73, 80, 81, 82, 89, 90, 91, 98, 99, 100, 107, 108, 109, 116, 117, 118, 125, 126, 127, 134, 135, 136

Offset: 1

dead

The b-file has been removed. - N. J. A. Sloane, Jan 19 2019

A016779 a(n) = (3*n + 1)^3.

Original entry on oeis.org

1, 64, 343, 1000, 2197, 4096, 6859, 10648, 15625, 21952, 29791, 39304, 50653, 64000, 79507, 97336, 117649, 140608, 166375, 195112, 226981, 262144, 300763, 343000, 389017, 438976, 493039, 551368, 614125, 681472, 753571, 830584, 912673, 1000000, 1092727, 1191016

Offset: 0

N. J. A. Sloane

The inverse binomial transform is 1, 63, 216, 162, 0, 0, 0 (0 continued). R. J. Mathar, May 07 2008

Perfect cubes with digital root 1 in base 10. Proof: perfect cubes are one of (3*s)^3, (3*s+1)^3 or (3*s+2)^3. Digital roots of (3*s)^3 are 0, digital roots of (3*s+1)^3 are 1, and digital roots of (3*s+2)^3 are 8, using trinomial expansion and the multiplicative property of digits roots. - R. J. Mathar, Jul 31 2010

			a(2) = (3*2+1)^3 = 343.
a(6) = (3*6+1)^3 = 6859.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.3.
Amarnath Murthy, Fabricating a perfect cube with a given valid digit sum (to be published)

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A016791, A054966, A226735.

[(3*n+1)^3: n in [0..30]]; // Vincenzo Librandi, May 09 2011

Table[(3n+1)^3,{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,64,343,1000},40] (* Harvey P. Dale, Oct 31 2016 *)

a(n)=(3*n+1)^3 \\ Charles R Greathouse IV, Jan 02 2012

Sum_{n>=0} 1/a(n) = 2*Pi^3 / (81*sqrt(3)) + 13*zeta(3)/27.
O.g.f.: (1 + 60*x + 93*x^2 + 8*x^3)/(1 - x)^4. - R. J. Mathar, May 07 2008
E.g.f.: (1 + 63*x + 108*x^2 + 27*x^3)*exp(x). - Ilya Gutkovskiy, Jun 16 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
Sum_{n>=1} (-1)^n/a(n) = A226735. - R. J. Mathar, Feb 07 2024

A016791 a(n) = (3*n + 2)^3.

Original entry on oeis.org

8, 125, 512, 1331, 2744, 4913, 8000, 12167, 17576, 24389, 32768, 42875, 54872, 68921, 85184, 103823, 125000, 148877, 175616, 205379, 238328, 274625, 314432, 357911, 405224, 456533, 512000, 571787, 636056, 704969, 778688, 857375, 941192, 1030301, 1124864, 1225043

Offset: 0

N. J. A. Sloane

Also the perfect cubes with digital root 8. [Proof: perfect cubes are either of the form (3n)^3 or of the form (3n+1)^3 or of the form (3n+2)^3. These subsets have digital root 9, 1 and 8 respectively.] - R. J. Mathar, Oct 02 2008

			a(4) = (3*4 + 2)^3 = 2744.
a(8) = (3*8 + 2)^3 = 17576.

Amarnath Murthy, Fabricating a perfect cube with a given valid digit sum (to be published)

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A054966, A016779, A016789, A073636.

(3*Range[0,40]+2)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{8,125,512,1331},40] (* Harvey P. Dale, Feb 20 2013 *)

a(n) = { (3*n + 2)^3 } \\ Harry J. Smith, Jul 18 2009

a(n) = A016789(n)^3. - Nathaniel Johnston, May 04 2011
G.f.: (8 + 93*x + 60*x^2 + x^3)/(1 - 4*x + 6*x^2 - 4*x^3 + x^4). - Colin Barker, Jan 02 2012
a(0)=8, a(1)=125, a(2)=512, a(3)=1331, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Feb 20 2013
Sum_{n>=0} 1/a(n) = -2*Pi^3 / (81*sqrt(3)) + 13*zeta(3)/27. - Amiram Eldar, Oct 02 2020

More terms from Harry J. Smith, Jul 18 2009

First digital root in proof in comment line corrected. - Ant King, May 01 2013

A159462 Numbers n with property that sod(n^3) = 5^3.

Original entry on oeis.org

341075, 423299, 446423, 542657, 638144, 661529, 667163, 786599, 798899, 828113, 837719, 841733, 842921, 861683, 869513, 879353, 883595, 887813, 887819, 905882, 912176, 912299, 919676, 923144, 927926, 928259, 928298, 943538, 950216, 954635

Offset: 1

Zak Seidov, Apr 12 2009

Numbers n with property that A007953(n^3) = 5^3.

			341075^3 = 39677989979796875, 3+9+6+7+7+9+8+9+9+7+9+7+9+6+8+7+5 = 125 = 5^3.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A054966 Numbers that are congruent to {0, 1, 8} mod 9. A054966 Possible sums of digits of cubes. A067075 a(n) = smallest number m such that the sum of the digits of m^3 is equal to n^3. A007953 Digital sum (i.e. sum of digits) of n. A159463 Numbers n with property that sod(n^3) = 6^3.

Select[Range[10^6],Total[IntegerDigits[#^3]]==125&] (* Harvey P. Dale, Jun 29 2022 *)

isok(n) = sumdigits(n^3) == 125; \\ Michel Marcus, Oct 16 2013

A159463 Numbers n with property that sod(n^3) = 6^3.

Original entry on oeis.org

3848163483, 4462569999, 4479677412, 4586158119, 4594661259, 4594665192, 4594700889, 4625720379, 4641588459, 5644008999, 5828410842, 5833034823, 5838252576, 5848025709, 6453471192, 6617331999, 6619097067, 6686657169, 7107126942, 7230291999, 7277907183

Offset: 1

Zak Seidov, Apr 12 2009

Numbers n with property that A007953(n^3) = 6^3.

			3848163483^3 = 56984998629886989599887999587, 5+6+9+8+4+9+9+8+6+2+9+8+8+6+9+8+9+5+9+9+8+8+7+9+9+9+5+8+7 = 216 = 6^3.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A054966 Numbers that are congruent to {0, 1, 8} mod 9. A054966 Possible sums of digits of cubes. A067075 a(n) = smallest number m such that the sum of the digits of m^3 is equal to n^3. A007953 Digital sum (i.e., sum of digits) of n.

Numbers n such that sum of digits of n^3 is k^3: A107679 (k=2), A290842 (k=3), A290843 (k=4), A159462 (k=5), this sequence (k=6).

a(16)-a(21) from Seiichi Manyama, Aug 12 2017

A067177 Cube root of A061096(n).

Original entry on oeis.org

1, 2, 3, 4, 14, 9, 13, 26, 27, 19, 53, 66, 76, 92, 132, 157, 353, 423, 559, 842, 927, 1192, 2129, 3846, 1966, 4289, 5826, 8782, 12116, 16299, 19129, 12599, 41013, 30355, 63413, 66942

Offset: 0

Amarnath Murthy, Jan 09 2002

R. J. Mathar, Table of n, a(n) for n = 0..51

Cf. A061096, A054966.

read("transforms"):
A067177 := proc(n)
    local ds,k ;
    ds := A054966(2+n) ;
    for k from 0 do
        if digsum(k^3) = ds then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Aug 28 2018

More terms from Sascha Kurz, Jan 28 2003

A275910 Numbers not congruent to 0, 1 or 8 mod 9.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 25, 29, 30, 31, 32, 33, 34, 38, 39, 40, 41, 42, 43, 47, 48, 49, 50, 51, 52, 56, 57, 58, 59, 60, 61, 65, 66, 67, 68, 69, 70, 74, 75, 76, 77, 78, 79, 83, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 97, 101, 102, 103, 104, 105, 106, 110, 111

Offset: 1

N. J. A. Sloane, Aug 26 2016

H. I. Okagbue, M. O. Adamu, S. A. Bishop and A. A. Opanuga, Properties of Sequences Generated by Summing the Digits of Cubed Positive Integers, Indian Journal Of Natural Sciences, Vol. 6 / Issue 32 / October 2015.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Complement of A054966.

DeleteCases[Range[120],?(MemberQ[{0,1,8},Mod[#,9]]&)] (* _Harvey P. Dale, Oct 08 2017 *)

Vec(x*(2+x+x^2+x^3+x^4+x^5+2*x^6)/((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)) + O(x^100)) \\ Colin Barker, Aug 26 2016

From Colin Barker, Aug 26 2016: (Start)
a(n) = a(n-1)+a(n-6)-a(n-7) for n>7.
G.f.: x*(2+x+x^2+x^3+x^4+x^5+2*x^6) / ((1-x)^2*(1+x)*(1-x+x^2)*(1+x+x^2)). (End)

cp's OEIS Frontend

A061096 Let k = n-th number that is a possible digit-sum for a cube (A054966); sequence gives smallest cube with digit-sum k.

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A135043 Duplicate of A054966.

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A016779 a(n) = (3*n + 1)^3.

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Magma

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A016791 a(n) = (3*n + 2)^3.

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A159462 Numbers n with property that sod(n^3) = 5^3.

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Mathematica

PARI

A159463 Numbers n with property that sod(n^3) = 6^3.

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A067177 Cube root of A061096(n).

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A275910 Numbers not congruent to 0, 1 or 8 mod 9.

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