A048766 -id:A048766 - cp's OEIS frontend

A347517 Partial sums of A347516.

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 38, 40, 41, 44, 45, 47, 49, 51, 52, 55, 56, 58, 60, 62, 63, 66, 67, 69, 71, 73, 74, 77, 78, 80, 82, 84, 85, 88, 89, 91, 93, 95, 96, 99, 100, 102, 104, 107, 108, 111, 112, 115, 117, 119, 120, 124, 125, 127, 129, 132, 133, 136

Offset: 1

Seiichi Manyama, Sep 04 2021

nonn

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

Cf. A006218, A048766, A094820, A347516.

f[n_] := DivisorSum[n, 1 &, # <= n^(1/3) &]; Accumulate @ Array[f, 100] (* Amiram Eldar, Sep 04 2021 *)

N=99; x='x+O('x^N); Vec(sum(k=1, N^(1/3), x^k^3/(1-x^k))/(1-x))

a(n) = sum(k=1, n, sumdiv(k, d, d^3<=k)); \\ Michel Marcus, Sep 05 2021

G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k^3)/(1 - x^k).

A109470 Sum of first n noncubes.

Original entry on oeis.org

2, 5, 9, 14, 20, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 201, 222, 244, 267, 291, 316, 342, 370, 399, 429, 460, 492, 525, 559, 594, 630, 667, 705, 744, 784, 825, 867, 910, 954, 999, 1045, 1092, 1140, 1189, 1239, 1290, 1342, 1395, 1449, 1504, 1560

Offset: 1

Jonathan Vos Post, Aug 28 2005

			a(6) = 2 + 3 + 4 + 5 + 6 + 7 = 27.
a(7) = 2 + 3 + 4 + 5 + 6 + 7 + 9 = 36.

Cf. A000537, A007412, A048766, A064524, A086849.

Accumulate[With[{no=60},Complement[Range[no],Range[Floor[Power[no, (3)^-1]]]^3]]]  (* Harvey P. Dale, Feb 14 2011 *)

a(n) = sum(i=1, n, i + sqrtnint(i + sqrtnint(i, 3), 3)); \\ Michel Marcus, Jun 20 2024

from sympy import integer_nthroot
def A109470(n): return ((m:=n+(k:=integer_nthroot(n,3)[0])+int(n>=(k+1)**3-k))*(m+1)>>1)-((r:=integer_nthroot(m,3)[0])*(r+1)>>1)**2 # Chai Wah Wu, Jun 17 2024

a(n) = Sum_{i=1..n} A007412(i).
a(n) = A000217(A007412(n)) - Sum_{i=1..floor((A007412(n)^(1/3)))} i^3.
a(n) = A000217(A007412(n)) - A000217(floor(A007412(n)^(1/3)))^2.
Let R = A007412(n) and S = floor(R^(1/3)); then a(n) = (R*(R+1))/2 - ((S*(S+1))/2)^2. - Gerald Hillier, Dec 21 2008

A113768 a(1) = 1, a(n+1) = a(n) + floor(a(n)^(1/3)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 133, 138, 143, 148, 153, 158, 163, 168, 173, 178, 183, 188, 193, 198, 203, 208, 213, 218, 224, 230, 236, 242, 248, 254, 260

Offset: 1

Jonathan Vos Post, Jan 19 2006

First 17 terms identical to A079645 (Integer part of the cube root of n divides n). Replacing cube root by square root gives A033638.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A033638, A048766, A079645.

[n le 1 select 1 else Self(n-1)+Floor(Self(n-1)^(1/3)): n in [1..75]]; // Vincenzo Librandi, Jul 29 2019

A[1]:= 1:
for n from 1 to 100 do A[n+1] := A[n] + floor(A[n]^(1/3)) od:
seq(A[i],i=1..100); # Robert Israel, Jul 28 2019

NestList[#+Floor[Surd[#,+3]]&,1,70] (* Harvey P. Dale, Jan 21 2013 *)

Conjecture: a(n) ~ (2/3)*n*sqrt((2/3)*n). - José María Grau Ribas, Feb 13 2024

Corrected and extended by Harvey P. Dale, Jan 21 2013

A132296 Sum of the noncube numbers less than or equal to n.

Original entry on oeis.org

0, 2, 5, 9, 14, 20, 27, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 201, 222, 244, 267, 291, 316, 342, 342, 370, 399, 429, 460, 492, 525, 559, 594, 630, 667, 705, 744, 784, 825, 867, 910, 954, 999, 1045, 1092, 1140, 1189, 1239, 1290, 1342, 1395, 1449

Offset: 1

Cino Hilliard, Nov 07 2007

nonn

			Let n=10. The sum of the noncube numbers <= 10 is 2+3+4+5+6+7+9+10 = 46, the 10th entry in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Accumulate[Table[If[IntegerQ[n^(1/3)],0,n],{n,60}]] (* Harvey P. Dale, Oct 16 2012 *)

g(n)=for(x=1,n,r=floor(x^(1/3));sumcu=(r*(r+1)/2)^2;sn=x*(x+1)/2;print1(sn-sumcu","))

from sympy import integer_nthroot
def A132296(n): return n*(n+1)-(((r:=integer_nthroot(n,3)[0])*(r+1))**2>>1)>>1 # Chai Wah Wu, Sep 03 2024

Let r = floor(n^(1/3)) = A048766(n). Then a(n) = n(n+1)/2 - (r(r+1)/2)^2 = A000217(n)-A000537(r).

A134918 Ceiling(n^(5/3)).

Original entry on oeis.org

1, 4, 7, 11, 15, 20, 26, 32, 39, 47, 55, 63, 72, 82, 92, 102, 113, 124, 136, 148, 160, 173, 187, 200, 214, 229, 243, 259, 274, 290, 306, 323, 340, 357, 375, 393, 411, 430, 449, 468, 488, 508, 528, 549, 570, 591, 613, 634, 657

Offset: 1

Mohammad K. Azarian, Nov 17 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A048766, A038605, A003059, A134914, A129011, A100196, A121536, A134917.

[Ceiling(n^(5/3)): n in [1..50]]; // Vincenzo Librandi, Feb 18 2013

Table[Ceiling[n^(5/3)], {n, 50}] (* Vincenzo Librandi, Feb 18 2013 *)

A187112 a(n) = cube root of the largest proper divisor of A187104(n).

Original entry on oeis.org

2, 3, 3, 4, 5, 5, 6, 5, 7, 8, 7, 9, 7, 10, 9, 7, 11, 12, 11, 13, 14, 13, 11, 15, 16, 11, 17, 15, 13, 18, 19, 11, 17, 13, 20, 21, 19, 22, 13, 23, 17, 24, 21, 13, 25, 19, 17, 26, 23, 27, 28, 25, 19, 29, 30, 17, 27, 31, 23, 17, 32, 33, 29, 19, 25, 34, 17, 23, 35, 19, 31, 36, 37, 33, 38, 19, 39, 29, 40, 35, 19, 23, 41, 42, 31, 37, 23, 43, 44, 29, 39, 45, 46, 41, 23, 47, 31, 35, 48, 23, 49, 43, 50

Offset: 1

Zak Seidov, Mar 05 2011

nonn

Zak Seidov, Table of n, a(n) for n = 1..2671

Cf. A187104 (largest proper divisor is a cube).

lista(nn) = {forcomposite(n=1, nn, if (ispower(divisors(n)[numdiv(n)-1], 3, &k), print1(k, ", ")););} \\ Michel Marcus, Aug 09 2014

a(n) = A048766(A032742(A187104(n))). - Michel Marcus, Aug 09 2014

A191291 The number of bases >=2 in which n is 3-digit number.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5

Offset: 1

Vladimir Shevelev, May 29 2011

Cf. A191279.

Table[Floor[Sqrt[n]]-Floor[CubeRoot[n]],{n,90}] (* Harvey P. Dale, Mar 22 2023 *)

a(n)=sqrtint(n)-floor((n+.5)^(1/3)) \\ Charles R Greathouse IV, Jan 03 2013

a(n) = floor(sqrt(n))-floor(n^(1/3)) = A000196(n) - A048766(n).

A258262 Cubes that are a sum of the cubes of three primes.

Original entry on oeis.org

128787625, 153130375, 356400829, 647214625, 1102302937, 1115157653, 1214767763, 1454419637, 3463512697, 14796346375, 18630700451, 21184951663, 21323063917, 21740999671, 24820429213, 29704593673, 32005984375, 38580208939, 51770001583, 53540005609, 68769820673, 74352915125, 89374579111, 94507253875, 113872553423

Offset: 1

Reinhard Zumkeller, Jun 13 2015

nonn

See comment in A258865.

			.   n |          a(n)
. ----+----------------------------------------------------
.   1 |     128787625 |   505^3 |  59^3 + 163^3 + 499^3
.   2 |     153130375 |   535^3 |  349^3 + 379^3 + 383^3
.   3 |     356400829 |   709^3 |  193^3 + 461^3 + 631^3
.   4 |     647214625 |   865^3 |  11^3 + 607^3 + 751^3
.   5 |    1102302937 |  1033^3 |  599^3 + 691^3 + 823^3
.   6 |    1115157653 |  1037^3 |  59^3 + 233^3 + 1033^3
.   7 |    1214767763 |  1067^3 |  97^3 + 269^3 + 1061^3
.   8 |    1454419637 |  1133^3 |  577^3 + 797^3 + 911^3
.   9 |    3463512697 |  1513^3 |  337^3 + 967^3 + 1361^3
.  10 |   14796346375 |  2455^3 |  1049^3 + 1789^3 + 1993^3
.  11 |   18630700451 |  2651^3 |  281^3 + 1889^3 + 2281^3
.  12 |   21184951663 |  2767^3 |  103^3 + 2179^3 + 2213^3
.  13 |   21323063917 |  2773^3 |  331^3 + 467^3 + 2767^3
.  14 |   21740999671 |  2791^3 |  769^3 + 1879^3 + 2447^3
.  15 |   24820429213 |  2917^3 |  31^3 + 1951^3 + 2591^3
.  16 |   29704593673 |  3097^3 |  1237^3 + 2081^3 + 2659^3
.  17 |   32005984375 |  3175^3 |  809^3 + 1789^3 + 2953^3
.  18 |   38580208939 |  3379^3 |  641^3 + 1993^3 + 3121^3
.  19 |   51770001583 |  3727^3 |  1399^3 + 1667^3 + 3541^3
.  20 |   53540005609 |  3769^3 |  11^3 + 1783^3 + 3631^3
.  21 |   68769820673 |  4097^3 |  1187^3 + 1861^3 + 3929^3
.  22 |   74352915125 |  4205^3 |  1657^3 + 1697^3 + 4019^3
.  23 |   89374579111 |  4471^3 |  1931^3 + 3163^3 + 3697^3
.  24 |   94507253875 |  4555^3 |  521^3 + 2833^3 + 4153^3
.  25 |  113872553423 |  4847^3 |  593^3 + 1237^3 + 4817^3  .

Chai Wah Wu, Table of n, a(n) for n = 1..367

Cf. A010057, A258865, A030078, A048766.

a258262 n = a258262_list !! (n-1)
a258262_list = filter ((== 1) . a010057) a258865_list

A263846 Floor of cube root of n-th prime.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 09 2015

nonn

Cf. A000006, A000196, A048766.

Floor[Surd[#,3]]&/@Prime[Range[120]] (* Harvey P. Dale, Jun 19 2022 *)

a(n) = sqrtnint(prime(n), 3); \\ Michel Marcus, Nov 10 2015

a(n) = A048766(A000040(n)). - Michel Marcus, Nov 10 2015

A277780 a(n) is the least k > n such that n*k^2 is a cube.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 27, 72, 80, 88, 96, 104, 112, 120, 54, 136, 144, 152, 160, 168, 176, 184, 81, 200, 208, 64, 224, 232, 240, 248, 108, 264, 272, 280, 288, 296, 304, 312, 135, 328, 336, 344, 352, 360, 368, 376, 162, 392, 400, 408, 416, 424, 128, 440

Offset: 1

Peter Kagey, Oct 30 2016

a(n) is bounded above by 8*n (A008590) because n*(8*n)^2 = (4*n)^3.
If and only if n is cubefree, a(n) = 8n. - David A. Corneth, Nov 01 2016
Theorem: If n = q*m^3 with q cubefree then k = q*(m+1)^3. - Hartmut F. W. Hoft, Nov 02 2016
Proof: let q have u distinct prime divisors p_i. Then q = Product_{i=1..u}(p_i^e_i) where e_i > 0 since p_i|q and e_i < 3 since q is cubefree. Therefore, e_i = 1 or e_i = 2. This yields q|k, i.e., q*t = k. Now for n*k^2 = q*m^3*q^2*t^2 = (q*m)^3 * t^2 to be a cube, t must be a cube. Now, k > n, so q*t/(q*m^3) = t/m^3. The least cube > m^3 is (m+1)^3 so k = q*(m+1)^3 which completes the proof. - David A. Corneth, Nov 03 2016

			a(24) = 81  because 24 *  81^2 =  54^3;
a(25) = 200 because 25 * 200^2 = 100^3;
a(26) = 208 because 26 * 208^2 = 104^3;
a(27) = 64  because 27 *  64^2 =  48^3.
The cubefree part of 144 is 18. The cubefull part of 144 is 8 = 2^3. Therefore, a(144) = 18 * 3^3 = 486. - _David A. Corneth_, Nov 01 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000578, A008590, A008834, A048766, A050985, A072905, A254767, A277781.

Cf. A002117, A013662, A013663, A013664.

Table[k = n + 1; While[! IntegerQ[(n k^2)^(1/3)], k++]; k, {n, 55}] (* Michael De Vlieger, Nov 04 2016 *)

a(n) = {my(k = n+1); while (!ispower(n*k^2, 3), k++); k;} \\ Michel Marcus, Oct 31 2016

a(n) = {my(f = factor(n)); f[, 2] = f[, 2]%3; f=factorback(f); n = sqrtnint(n/f,3); (n+1)^3 * f} \\ David A. Corneth, Nov 01 2016

a(n) = A050985(n) * A000578(1+A048766(A008834(n))). [Formula given in comments expressed with A-numbers] - Antti Karttunen, Nov 02 2016.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + (3*zeta(4) + 3*zeta(5) + zeta(6))/zeta(3) = 7.13539675963975495073... . - Amiram Eldar, Feb 17 2024

cp's OEIS Frontend

A347517 Partial sums of A347516.

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A109470 Sum of first n noncubes.

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A113768 a(1) = 1, a(n+1) = a(n) + floor(a(n)^(1/3)).

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A132296 Sum of the noncube numbers less than or equal to n.

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A134918 Ceiling(n^(5/3)).

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A187112 a(n) = cube root of the largest proper divisor of A187104(n).

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A191291 The number of bases >=2 in which n is 3-digit number.

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A258262 Cubes that are a sum of the cubes of three primes.

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