A016755 -id:A016755 - cp's OEIS frontend

A381824 Odd cubefull numbers: odd numbers that are divisible by the cube of any of their prime factors.

1, 27, 81, 125, 243, 343, 625, 729, 1331, 2187, 2197, 2401, 3125, 3375, 4913, 6561, 6859, 9261, 10125, 12167, 14641, 15625, 16807, 16875, 19683, 24389, 27783, 28561, 29791, 30375, 35937, 42875, 50625, 50653, 59049, 59319, 64827, 68921, 78125, 79507, 83349, 83521, 84375, 91125

Offset: 1

Amiram Eldar, Mar 08 2025

Numbers whose prime factorization has primes and exponents that are larger than 2 (except for 1 whose prime factorization is empty).

Numbers k such that A020639(k) >= 3 and A051904(k) >= 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A036966.
Subsequences: A016755 (odd cubes), A381825 (odd cubefull exponentially odd numbers).
Cf. A020639, A051904, A065483.

Join[{1}, Select[Range[3, 10000, 2], Min[FactorInteger[#][[;; , 2]]] > 2 &]]

isok(k) = k == 1 || (k % 2 && vecmin(factor(k)[, 2]) > 2);

Sum_{n>=1} 1/a(n) = Product_{p prime >= 3} (1 + 1/(p^2*(p-1))) = (4/5) * A065483 = 1.07182732285947779727... .

A017115 a(n) = (8*n + 4)^3.

Original entry on oeis.org

64, 1728, 8000, 21952, 46656, 85184, 140608, 216000, 314432, 438976, 592704, 778688, 1000000, 1259712, 1560896, 1906624, 2299968, 2744000, 3241792, 3796416, 4410944, 5088448, 5832000, 6644672, 7529536, 8489664, 9528128, 10648000, 11852352, 13144256, 14526784, 16003008

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002117, A016755, A016827, A017113.

[(8*n+4)^3: n in [0..35] ]; // Vincenzo Librandi, Jul 21 2011

LinearRecurrence[{4, -6, 4, -1},{64, 1728, 8000, 21952},24] (* Ray Chandler, Aug 04 2015 *)

G.f.: 64*(1+x)*(x^2 + 22*x + 1)/(x-1)^4. - R. J. Mathar, Jul 14 2016
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^3.
a(n) = 2^3*A016827(n) = 2^6*A016755(n).
Sum_{n>=0} 1/a(n) = 7*zeta(3)/512.
Sum_{n>=0} (-1)^n/a(n) = Pi^3/2048. (End)
E.g.f.: 64*exp(x)*(1 + 26*x + 36*x^2 + 8*x^3). - Stefano Spezia, May 27 2025

A017331 a(n) = (10*n + 5)^3.

Original entry on oeis.org

125, 3375, 15625, 42875, 91125, 166375, 274625, 421875, 614125, 857375, 1157625, 1520875, 1953125, 2460375, 3048625, 3723875, 4492125, 5359375, 6331625, 7414875, 8615125, 9938375, 11390625, 12977875, 14706125, 16581375, 18609625, 20796875, 23149125, 25672375

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002117, A016755, A017329.

[(10*n+5)^3: n in [0..35]]; // Vincenzo Librandi, Aug 02 2011

Table[(10*n + 5)^3, {n, 0, 30}] (* Amiram Eldar, Apr 18 2023 *)
LinearRecurrence[{4,-6,4,-1},{125,3375,15625,42875},30] (* Harvey P. Dale, Aug 23 2024 *)

a(n)=(10*n+5)^3 \\ Charles R Greathouse IV, Aug 02 2011

G.f.: 125*(x+1)*(x^2 + 22*x + 1)/(x-1)^4. - Colin Barker, Nov 14 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^3.
a(n) = 5^3 * A016755(n).
Sum_{n>=0} 1/a(n) = 7*zeta(3)/1000.
Sum_{n>=0} (-1)^n/a(n) = Pi^3/4000. (End)

A239065 n^3*(n^4 + n^2 - 1).

Original entry on oeis.org

1, 152, 2403, 17344, 81125, 287496, 840007, 2129408, 4841289, 10099000, 19646891, 36078912, 63117613, 105948584, 171615375, 269479936, 411753617, 614103768, 896340979, 1283192000, 1805163381, 2499500872, 3411249623, 4594420224, 6113265625, 8043673976

Offset: 1

Philippe Deléham, Mar 09 2014

Row sums of A016755 read as triangular array.

			A016755, as triangular array begins:
1;
27, 125;
343, 729, 1331;
2197, 3375, 4913, 6859;
9261, 12167, 15625, 19683, 24389;
29791, 35937, 42875, 50653, 59319, 68921;..
Row sums are:
1;
3^3 + 5^3 = 27 + 125 = 152;
7^3 + 9^3 + 11^3 = 343 + 729 + 1331 = 2403;
13^3 + 15^3 + 17^3 + 19^3 = 2197 + 3375 + 4913 + 6859 = 17344;
21^3 + 23^3 + 25^3 + 27^3 + 29^3 = 9261 + 12167 + 15625 + 19683 + 24389 = 81125;
31^3 + 33^3 + 35^3 + 37^3 + 39^3 + 41^3 = 287496 = 66^3.

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

Cf. A016755.

[n^3*(n^4 + n^2 - 1): n in [1..30]]; // Vincenzo Librandi, Mar 11 2014

A239065:=n->n^7 + n^5 - n^3; seq(A239065(n), n=1..30); # Wesley Ivan Hurt, Mar 09 2014

Table[n^7 + n^5 - n^3, {n, 30}] (* Wesley Ivan Hurt, Mar 09 2014 *)
CoefficientList[Series[(1 + 144 x + 1215 x^2 + 2320 x^3 + 1215 x^4 + 144 x^5 + x^6)/(x - 1)^8, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 11 2014 *)

a(n) = n^7+n^5-n^3 \\ Charles R Greathouse IV, Mar 09 2014

a(n) = n^7 + n^5 - n^3.

G.f.: x*(1+144*x+1215*x^2+2320*x^3+1215*x^4+144*x^5+x^6)/(x-1)^8.

A280865 Expansion of 1/(1 - Sum_{k>=0} x^((2*k+1)^3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 30, 33, 37, 42, 48, 55, 63, 72, 82, 93, 105, 118, 132, 147, 163, 180, 198, 217, 237, 258, 280, 303, 327, 352, 378, 405, 433, 463, 496

Offset: 0

Ilya Gutkovskiy, Jan 09 2017

nonn

Number of compositions (ordered partitions) of n into odd cubes (A016755).

			a(28) = 3 because we have [27, 1], [1, 27] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000578, A016755, A023358, A003108, A078128, A279329, A280130.

nmax = 82; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)^3, {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=0} x^((2*k+1)^3)).

A330414 Cyclops primes that become a cube when the middle "0" is removed.

Original entry on oeis.org

68059, 1170649, 4560533, 7530571, 136501919, 158103251, 173703979, 212503933, 226605187, 356101289, 362604691, 382702753, 439806977, 518905117, 811802737, 954403993, 19484041249, 19956016979, 22635071297, 24658046551, 27263097773, 34635012697, 35326042667, 37166072149, 39668022287, 41499095543, 44839062449

Offset: 1

Rodolfo Ruiz-Huidobro, Dec 14 2019

			a(1) = 68059 because 6859 = 19^3 is the first cube that results from the removal of the 0 digit from a cyclops prime.
136501919 is a term because 13651919 is 239^3.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 60 terms from Rodolfo Ruiz-Huidobro)

Cf. A329737, A134809, A016755.

count:= 0: Res:= NULL:
for d from 2 to 6 do
  for n from ceil(10^((2*d-1)/3)) to floor((10^(2*d)-1)^(1/3)) do
    L:=convert(n^3,base,10);
    if member(0,L) then next fi;
    a:= n^3 mod 10^d;
    p:= 10*(n^3-a)+a;
    if isprime(p) then
      count:= count+1; Res:= Res, p;
    fi
od od:
Res; # Robert Israel, Dec 24 2019

seq(n)={my(i=0, L=List()); while(#Lt==0,v), my(m=fromdigits(concat([v[1..k], 0, v[k+1..#v]]))); if(isprime(m), listput(L,m)))); Vec(L)} \\ Andrew Howroyd, Dec 20 2019

A338447 Sums of consecutive odd positive cubes.

Original entry on oeis.org

1, 27, 28, 125, 152, 153, 343, 468, 495, 496, 729, 1072, 1197, 1224, 1225, 1331, 2060, 2197, 2403, 2528, 2555, 2556, 3375, 3528, 4257, 4600, 4725, 4752, 4753, 4913, 5572, 6859, 6903, 7632, 7975, 8100, 8127, 8128, 8288, 9261, 10485, 11772, 11816, 12167, 12545, 12888, 13013

Offset: 1

Ilya Gutkovskiy, Oct 28 2020

nonn

			495 is in the sequence because 495 = 3^3 + 5^3 + 7^3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of cubes

Cf. A002593, A016755, A042965, A217843, A265845, A290276, A320728.

lista(nn) = {my(list = List()); forstep (i=1, nn, 2, my(s = 0); forstep(j=i, 1, -2, s += j^3; if (s > nn^3, break); listput(list, s););); Set(list);} \\ Michel Marcus, Nov 13 2020

A371835 Triangle read by rows: T(n,k) is the number of points (x,y,z) satisfying |x|+|y|+|z|<=n and max(|x|,|y|,|z|)<=k; 0<=k<=n.

Original entry on oeis.org

1, 1, 7, 1, 19, 25, 1, 27, 57, 63, 1, 27, 93, 123, 129, 1, 27, 117, 195, 225, 231, 1, 27, 125, 263, 341, 371, 377, 1, 27, 125, 311, 461, 539, 569, 575, 1, 27, 125, 335, 569, 719, 797, 827, 833, 1, 27, 125, 343, 649, 895, 1045, 1123, 1153, 1159

Offset: 0

Peter Kagey, Apr 07 2024

For all pairs of positive integers (a,b), T(a*m,b*m) satisfies a cubic polynomial in m.

			Table begins:
  n\k| 0  1   2   3   4    5    6    7    8    9   10
  ---+-----------------------------------------------
   0 | 1
   1 | 1  7
   2 | 1 19  25
   3 | 1 27  57  63
   4 | 1 27  93 123 129
   5 | 1 27 117 195 225  231
   6 | 1 27 125 263 341  371  377
   7 | 1 27 125 311 461  539  569  575
   8 | 1 27 125 335 569  719  797  827  833
   9 | 1 27 125 343 649  895 1045 1123 1153 1159
  10 | 1 27 125 343 697 1051 1297 1447 1525 1555 1561

Sela Fried, On a problem related to the integer lattice and its layers, 2024.
Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024. See p. 5.

T(n,k)   =    8*n^3 +  12*n^2 +  6*n + 1    = A016755(k) if k <= n/3.
T(m,m)   =   (4*n^3 +   6*n^2 +  8*n + 3)/3 = A001845(m).
T(2m,m)  =  (20*n^3 +  24*n^2 + 10*n + 3)/3 = A371532(m).
T(3m,2m) =   32*n^3 +  18*n^2 +  6*n + 1    = A371515(m).
T(4m,3m) = (244*n^3 +  96*n^2 + 26*n + 3)/3.
T(5m,2m) = (188*m^3 + 132*m^2 + 28*m + 3)/3.
T(5m,3m) = (404*m^3 + 150*m^2 + 28*m + 3)/3.
T(5m,4m) = (488*m^3 + 150*m^2 + 34*m + 3)/3.
Conjectures:
T(n,k) = (-84*k^3 + 108*k^2*n - 72*k^2 - 36*k*n^2 + 72*k*n - 6*k + 4*n^3 - 12*n^2 + 8*n + 3)/3 for (n-2)/3 <= k <= n/2.
T(n,k) = (12*k^3 - 36*k^2*n + 36*k*n^2 + 6*k - 8*n^3 + 6*n^2 + 2*n + 3)/3 for (n-1)/2 <= k <= n.
The two conjectures are true. See links. - Sela Fried, Jul 05 2024

A385029 a(n) = Sum_{-n <= a, b, c <= n} (b^2 - 4ac).

Original entry on oeis.org

18, 250, 1372, 4860, 13310, 30758, 63000, 117912, 205770, 339570, 535348, 812500, 1194102, 1707230, 2383280, 3258288, 4373250, 5774442, 7513740, 9648940, 12244078, 15369750, 19103432, 23529800, 28741050, 34837218, 41926500, 50125572, 59559910, 70364110, 82682208, 96668000

Offset: 1

Darío Clavijo, Jun 15 2025

There are (2*n + 1)^3 combinations of a, b, c.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000384, A002378, A014105, A016755, A016754, A055112, A384666.

A385029[n_] := (n*(n + 1)*(2*n + 1)^3)/3;
Array[A385029, 50] (* Paolo Xausa, Jun 18 2025 *)

a = lambda n: ((n*n+n)*((n << 1)+1)**3)//3
print([a(n) for n in range(1, 11)])

a(n) = (n*(n+1)*(2*n+1)^3)/3.
a(n) = (A055112(n)*A016754(n))/3.
a(n) = (A002378(n)*A016755(n))/3.
G.f.: 2*x*(9 + 71*x + 71*x^2 + 9*x^3)/(1 - x)^6. - Stefano Spezia, Jun 15 2025
From Amiram Eldar, Jun 18 2025; (Start)
Sum_{n>=1} 1/a(n) = 21*(1 - zeta(3)/2) - 12*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi^3/8 + 3*Pi - 21. (End)

A303743 a(n) is a number of lattice points in 3D Cartesian grid between cube with edge length 2*n centered in origin and its inscribed sphere. Three pairs of the cube's faces are parallel to the planes XOY, XOZ, YOZ respectively.

Original entry on oeis.org

0, 0, 8, 92, 220, 412, 784, 1272, 1848, 2696, 3692, 5020, 6460, 8176, 10248, 12720, 15464, 18476, 21988, 25924, 30016, 35040, 40248, 46052, 52388, 59132, 66364, 74416, 83256, 92304, 102500, 112988, 124076, 136252, 148936, 162648, 176928, 192332, 208100, 225284, 243088

Offset: 1

Kirill Ustyantsev, Apr 29 2018

nonn

If two parallel faces of the inscribed cube are parallel XOY-plane and other two pairs are parallel planes x=y and x=-y respectively we'll have another sequence.

			For n=3 we have 8 points between the defined cube and its inscribed sphere:
  (-2,-2,-2)
  (-2,-2, 2)
  (-2, 2,-2)
  (-2, 2, 2)
  ( 2,-2,-2)
  ( 2,-2, 2)
  ( 2, 2,-2)
  ( 2, 2, 2)

Cf. A000605, A016755.

For the 2D case see A303642.

a(n) = sum(x=-n+1, n-1, sum(y=-n+1, n-1, sum(z=-n+1, n-1, x*x+y*y+z*z>n^2))); \\ Michel Marcus, Jun 23 2018

for n in range (1, 42):
  count=0
  n2 = n*n
  for x in range(-n+1, n):
    for y in range(-n+1, n):
      for z in range(-n+1, n):
        if x*x+y*y+z*z > n2:
          count += 1
  print(count)

a(n) = A016755(n-1) - A000605(n) - 6.

cp's OEIS Frontend

A381824 Odd cubefull numbers: odd numbers that are divisible by the cube of any of their prime factors.

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A017115 a(n) = (8*n + 4)^3.

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A017331 a(n) = (10*n + 5)^3.

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A239065 n^3*(n^4 + n^2 - 1).

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A280865 Expansion of 1/(1 - Sum_{k>=0} x^((2*k+1)^3)).

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A330414 Cyclops primes that become a cube when the middle "0" is removed.

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A338447 Sums of consecutive odd positive cubes.

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PARI

A371835 Triangle read by rows: T(n,k) is the number of points (x,y,z) satisfying |x|+|y|+|z|<=n and max(|x|,|y|,|z|)<=k; 0<=k<=n.

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A385029 a(n) = Sum_{-n <= a, b, c <= n} (b^2 - 4ac).