A068601 -id:A068601 - cp's OEIS frontend

A240975 The number of distinct prime factors of n^3-1.

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

Offset: 1

R. J. Mathar, Aug 05 2014

			3^3-1 = 26 = 2*13, so a(3) = 2.
0 has no prime factors, so a(1) = 0.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A001221, A068601, A082863, A245909.

A240975 := proc(n)
    A001221(n^3-1) ;
end proc:

a[n_] := PrimeNu[n^3-1]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Sep 13 2024 *)

a(n) = if(n<=1,0,omega(n^3-1)); \\ Joerg Arndt, Aug 06 2014

from sympy import primefactors
def A240975(n):
    return len(primefactors(n**3-1)) # Chai Wah Wu, Aug 06 2014

a(prime(n)) = A245909(n).

a(n) = A001221(A068601(n)) for n >= 2. - Michel Marcus, Aug 06 2014

A268861 Cubefree numbers n such that n + 1 is a perfect cube.

Original entry on oeis.org

7, 26, 63, 124, 215, 342, 511, 1330, 1727, 2196, 2743, 3374, 4095, 7999, 9260, 10647, 12166, 13823, 17575, 19682, 24388, 26999, 29790, 32767, 39303, 42874, 46655, 54871, 59318, 63999, 74087, 79506, 85183, 91124, 103822, 110591, 124999, 132650, 140607, 148876

Offset: 1

K. D. Bajpai, Feb 14 2016

nonn

Intersection of A004709 and A068601. - Michel Marcus, Feb 15 2016

			a(2) = 26 = 2 * 13 that is cubefree. 26 + 1 = 27 = 3^3 (perfect cube).
a(4) = 124 = 2 * 2 * 31 that is cubefree. 124 + 1 = 125 = 5^3 (perfect cube).

K. D. Bajpai, Table of n, a(n) for n = 1..400

Cf. A004709, A068601, A121628, A221793, A268752.

cubefree:= proc(n) local t;
  max(seq(t[2],t=ifactors(n)[2])) <= 2
end proc:
select(cubefree, [seq(i^3-1,i=2..100)]); # Robert Israel, Mar 03 2016

Select[Range[150000], FreeQ[FactorInteger[#], {, k /; k > 2}] && IntegerQ[CubeRoot[# + 1]] &]
Select[Range[2,70]^3,Max[FactorInteger[#-1][[All,2]]]<3&]-1 (* Harvey P. Dale, Oct 11 2021 *)

for(n=1, 1e5, f = factor(n)[, 2]; if((#f == 0) || vecmax(f) < 3, if(ispower(n + 1, 3), print1(n, ", "))));

A271717 Integers k such that both k and k^3-1 are the sum of two positive cubes (see A003325).

Original entry on oeis.org

9, 11664, 36864, 38134, 345744, 1750329, 4782969, 20820969, 47775744, 65804544, 95004009, 150994944, 448084224, 733055625, 1093955625, 1416167424, 2197265625, 4318066944, 5194805625, 6198727824, 7169347584, 10771948944, 13013105625, 19591041024, 32427005625

Offset: 1

Altug Alkan, Apr 12 2016

nonn

Values of a^3 + b^3 such that (a^3 + b^3)^3 - 1 is of the form x^3 + y^3 where a, b, x, y > 0.
38134 = 2*23*829 is the first term that is nonsquare. What are the next square terms of this sequence?
n is a member of A007412 and n^3 is a member of A003072, obviously.

			9 is a term because 9 = 1^3 + 2^3 and 9^3 - 1 = 6^3 + 8^3.

Chai Wah Wu, Table of n, a(n) for n = 1..33

Cf. A003325, A050787, A068601.

isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
for(n=1, 1e7, if(isA003325(n) && isA003325(n^3-1), print1(n, ", ")));

a(8)-a(16) from Chai Wah Wu, Apr 17 2016

a(17)-a(25) from Chai Wah Wu, Jul 21 2025

A274578 Nonsquare k such that k^3 - 1 is the average of two positive cubes.

Original entry on oeis.org

2305, 2629, 4117, 7060, 37444, 46081, 113320, 208545, 449569, 474553, 507325, 1224757, 1499068, 1927405, 1931077, 2263129, 2350909, 2447596, 3107841, 4065517, 4274932, 4303321, 5646685, 6582865, 7225597, 10386273, 18432001, 21936709, 24218425, 24362989, 27351417

Offset: 1

Altug Alkan, Jun 29 2016

nonn

The equation x^3 + y^3 = 2*z^3 has no integer solution triple (x, y, z) for x > y and z is nonzero. So this sequence focuses on the equation x^3 + y^3 = 2*(z^3 - 1) where x, y > 0.

			2305 is a term because it is not a square and 2305^3 - 1 = (144^3 + 2904^3) / 2.

Chai Wah Wu, Table of n, a(n) for n = 1..66

Cf. A000037, A003325, A068601, A273822.

isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
lista(nn) = for(n=1, nn, if(isA003325(2*(n^3-1)) && !issquare(n), print1(n, ", ")));

a(9)-a(25) from Chai Wah Wu, Aug 07 2020

a(26)-a(31) from Chai Wah Wu, Jun 30 2025

A362496 Square array A(n, k), n, k >= 0, read by upwards antidiagonals; if Newton's method applied to the complex function f(z) = z^3 - 1 and starting from n + ki reaches or converges to exp(2riPi/3) for some r in 0..2, then A(n, k) = r, otherwise A(n, k) = -1 (where i denotes the imaginary unit).

Original entry on oeis.org

-1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1

Offset: 0

Rémy Sigrist, Apr 22 2023

This sequence is related to the Newton fractal, and exhibits similar rich patterns (see illustration in Links section).

			Array A(n, k) begins:
  n\k |  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
  ----+------------------------------------------------------
    0 | -1  1  1  1  1  1  1  1  1  1   1   1   1   1   1   1
    1 |  0  0  0  1  1  1  1  1  1  1   1   1   1   1   1   1
    2 |  0  0  0  0  2  1  1  1  1  1   1   1   1   1   1   1
    3 |  0  0  0  0  0  2  2  1  1  1   1   1   1   1   1   1
    4 |  0  0  0  0  0  0  1  2  2  1   2   1   1   1   1   1
    5 |  0  0  0  0  0  0  0  0  2  2   0   1   1   1   1   1
    6 |  0  0  0  0  0  0  0  0  2  1   0   2   2   1   2   2
    7 |  0  0  0  0  0  0  0  0  0  0   0   2   2   2   1   0
    8 |  0  0  0  0  0  0  0  0  0  0   0   1   2   2   2   0
    9 |  0  0  0  0  0  0  0  0  0  0   0   0   2   1   1   0
   10 |  0  0  0  0  0  0  0  0  0  0   0   0   0   2   0   0
   11 |  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0
   12 |  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0
   13 |  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0
   14 |  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0
   15 |  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0

Rémy Sigrist, Colored representation of the square array for n, k <= 1000 (black, white, blue and red pixels denote, respectively, -1, 0, 1 and 2)
Rémy Sigrist, PARI program
Wikipedia, Newton fractal
Wikipedia, Newton's method

Cf. A068601.

PARI
```
See Links section.
```

A062841 Palindromes of the form k^3-1.

Original entry on oeis.org

0, 7, 999, 999999, 258474852, 999999999, 999999999999, 999999999999999, 999999999999999999, 999999999999999999999, 999999999999999999999999, 999999999999999999999999999, 999999999999999999999999999999, 999999999999999999999999999999999

Offset: 1

Erich Friedman, Jul 21 2001

Sequence is infinite as (10^k)^3-1 is a term for all k >= 0. - Michael S. Branicky, Mar 27 2021

			999 = 10^3-1 and is a palindrome.

Intersection of A002113 and A068601.

For[n=0,n<100000000,n++,If[n^3-1==IntegerReverse[n^3-1],Print[n^3-1]]] (* Dylan Delgado, Mar 02 2021 *)
Select[Range[10^7]^3-1,PalindromeQ] (* The program generates the first ten terms of the sequence. *) (* Harvey P. Dale, Oct 08 2023 *)

def afind(limit):
  for n in range(limit+1):
    s = str(n**3 - 1)
    if s == s[::-1]: print(int(s), end=", ")
print(afind(10**7)) # Michael S. Branicky, Mar 27 2021

One more term from Emeric Deutsch, Feb 26 2005

a(10)-a(14) from Michael S. Branicky, Mar 27 2021

A117197 a(n) = (n^3 - 1)^3.

Original entry on oeis.org

0, 343, 17576, 250047, 1906624, 9938375, 40001688, 133432831, 385828352, 997002999, 2352637000, 5150827583, 10590025536, 20638466407, 38409197624, 68669157375, 118515478528, 198257271191, 322546580712, 511808023999

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 21 2006

Cubes a(1-1),a(8-1),a(27-1),a(64-1),...

Cf. A000578 (n^3), A068601 (n^3-1).

(Range[20]^3-1)^3 (* Harvey P. Dale, Mar 25 2013 *)

a(n) = n^9 - 3n^6 + 3n^3 - 1.
G.f.: (x^9-2*x^8+694*x^7+14902*x^6+87160*x^5+155914*x^4+89722*x^3+14146*x^2+ 343*x)/(x-1)^10. - Harvey P. Dale, Mar 25 2013 [corrected by Georg Fischer, May 15 2019]
a(n) = A000578(A068601(n)). - Michel Marcus, May 15 2019

Edited and corrected by Franklin T. Adams-Watters, Apr 26 2006

A336194 Table read by antidiagonals upwards: T(n,k) = (n - 1)*k^3 - 1, with n > 1 and k > 0.

Original entry on oeis.org

0, 1, 7, 2, 15, 26, 3, 23, 53, 63, 4, 31, 80, 127, 124, 5, 39, 107, 191, 249, 215, 6, 47, 134, 255, 374, 431, 342, 7, 55, 161, 319, 499, 647, 685, 511, 8, 63, 188, 383, 624, 863, 1028, 1023, 728, 9, 71, 215, 447, 749, 1079, 1371, 1535, 1457, 999, 10, 79, 242, 511, 874, 1295, 1714, 2047, 2186, 1999, 1330

Offset: 2

Stefano Spezia, Jul 11 2020

T(n, k) is a sharp upper bound of the tree width of a graph G that does not contain a clique on n vertices nor a minimal separator of size larger than k (see Theorem 2.1 in Pilipczuk et al.).

All the square matrices starting at top left of the table T are singular except for the 2 X 2 submatrix: det([0, 7; 1, 15]) = -7.

			The table starts at row n = 2 and column k = 1 as:
0   7   26   63  124   215 ...
1  15   53  127  249   431 ...
2  23   80  191  374   647 ...
3  31  107  255  499   863 ...
4  39  134  319  624  1079 ...
5  47  161  383  749  1295 ...
...

Marcin Pilipczuk, Ni Luh Dewi Sintiari, Stéphan Thomassé and Nicolas Trotignon, (Theta, triangle)-free and (even hole, K4)-free graphs. Part 2 : bounds on treewidth, arXiv:2001.01607 [cs.DM], 2020. See p. 7.
Index entries for sequences related to trees.

Cf. A000578, A001093, A001477 (k = 1), A004771 (k = 2), A068601 (n = 2), A085537, A109129, A123865 (main diagonal), A325543, A325612.

T[n_,k_]:=(n-1)*k^3-1; Flatten[Table[T[n+1-k,k],{n,2,12},{k,1,n-1}]]

PARI
```
T(n, k) = (n - 1)*k^3 - 1
```

O.g.f.: x^2*y*(y*(7 - 2*y + y^2) + x*(1 - y)^3)/((1 - x)^2*(1 - y)^4).
E.g.f.: -1 + exp(x) - x + exp(y)*x + exp(y)*(1 + y + 3*y^2 + y^3) + exp(x + y)*(-1 +(-1 + x)*y*(1 + 3*y + y^2)).
T(n, k) = n*A000578(k) - A001093(k).
T(n, n) = A085537(n) - 1 for n > 1.
T(n, k) = T(n+1, 1)*T(2, k) + T(n, 1).

A369181 a(n) = (n^3-1)!.

Original entry on oeis.org

1, 5040, 403291461126605635584000000, 1982608315404440064116146708361898137544773690227268628106279599612729753600000000000000

Offset: 1

Karol A. Penson, Jan 15 2024

nonn

Cf. A000142, A068601, A256511.

Maple
```
seq((n^3-1)! , n = 1..4);
```

a(n) = (2*Pi)^(1/2 - n^2/2)*(n^2)^(n^3 - 1/2)*Product_{j=0..n^2-1} Gamma(n + j/n^2).

a(n) = A000142(A068601(n)).

cp's OEIS Frontend

A240975 The number of distinct prime factors of n^3-1.

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A268861 Cubefree numbers n such that n + 1 is a perfect cube.

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A271717 Integers k such that both k and k^3-1 are the sum of two positive cubes (see A003325).

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A274578 Nonsquare k such that k^3 - 1 is the average of two positive cubes.

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A062841 Palindromes of the form k^3-1.

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Python

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A117197 a(n) = (n^3 - 1)^3.

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A336194 Table read by antidiagonals upwards: T(n,k) = (n - 1)*k^3 - 1, with n > 1 and k > 0.

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