A167222 -id:A167222 - cp's OEIS frontend

A077121 Number of integer squares <= n^3.

1, 2, 3, 6, 9, 12, 15, 19, 23, 28, 32, 37, 42, 47, 53, 59, 65, 71, 77, 83, 90, 97, 104, 111, 118, 126, 133, 141, 149, 157, 165, 173, 182, 190, 199, 208, 217, 226, 235, 244, 253, 263, 273, 282, 292, 302, 312, 323, 333, 344, 354, 365, 375, 386, 397, 408, 420

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) = number of terms in n-th row of A167222. - Reinhard Zumkeller, Oct 31 2009

			Squares <= 3^3 = 27: 0, 1, 4, 9, 16 and 25, hence a(3) = 6.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000093, A065733, A077113, A167222, A167223.

a[n_] := Floor[Sqrt[n^3]] + 1; Array[a, 100, 0] (* Amiram Eldar, Apr 06 2025 *)

from math import isqrt
def A077121(n): return isqrt(n**3)+1 # Chai Wah Wu, Sep 08 2024

a(n) = floor(n^(3/2))+1 = A000093(n) + 1.

A167224 Table of primes of the form n^3 - k^2, 0<=k<=A077121(n).

Original entry on oeis.org

7, 23, 11, 2, 109, 89, 61, 191, 167, 47, 307, 199, 19, 503, 487, 463, 431, 223, 151, 71, 53, 991, 919, 271, 1327, 1231, 1187, 547, 431, 307, 1607, 1559, 1439, 1367, 1103, 887, 503, 359, 47, 2161, 2053, 1873, 1621, 1297, 433, 2719, 2663, 2383, 1783, 1223, 1063

Offset: 1

Reinhard Zumkeller, Oct 31 2009

Primes in A167222;

A161681 is the range of this table.

			7;23,11,2;;109,89,61;191,167,47;307,199,19;503,487,... .

R. Zumkeller, Table of n, a(n) for n = 1..10000
R. Zumkeller, Some Examples

A261893 a(n) = (n+1)^3 - n^2.

Original entry on oeis.org

1, 7, 23, 55, 109, 191, 307, 463, 665, 919, 1231, 1607, 2053, 2575, 3179, 3871, 4657, 5543, 6535, 7639, 8861, 10207, 11683, 13295, 15049, 16951, 19007, 21223, 23605, 26159, 28891, 31807, 34913, 38215, 41719, 45431, 49357, 53503, 57875, 62479, 67321, 72407

Offset: 0

Reinhard Zumkeller, Sep 05 2015

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

Cf. A000290, A000578.

Subsequence of A167222.

a261893 n = n * (n * (n + 2) + 3) + 1
a261893_list = zipWith (-) (tail a000578_list) a000290_list

[n^3+2*n^2+3*n+1: n in [0..50]]; // Bruno Berselli, Jul 04 2016

Table[n^3 + 2 n^2 + 3 n + 1, {n, 0, 50}] (* Bruno Berselli, Jul 04 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,7,23,55},50] (* Harvey P. Dale, Mar 01 2023 *)

makelist(n^3+2*n^2+3*n+1, n, 0, 50); /* Bruno Berselli, Jul 04 2016 */

vector(50, n, n--; n^3+2*n^2+3*n+1) \\ Bruno Berselli, Jul 04 2016

[n^3+2*n^2+3*n+1 for n in range(50)]; # Bruno Berselli, Jul 04 2016

a(n) = n^3 + 2*n^2 + 3*n + 1.
a(n) = A000578(n+1) - A000290(n).
O.g.f.: (1 + 3*x + x^2 + x^3)/(1 - x)^4. - Bruno Berselli, Jul 04 2016
E.g.f.: (1 + 6*x + 5*x^2 + x^3)*exp(x). - Bruno Berselli, Jul 04 2016

A373218 Cubes equal to the sum of a factorial number and a square.

Original entry on oeis.org

1, 27, 729, 46656

Offset: 1

Gonzalo Martínez, May 28 2024

Cubes c^3 for which the c-th row of A167222 includes a factorial number. - Michel Marcus, May 28 2024
a(5) > 8*10^24. - Michael S. Branicky, Jun 23 2024
a(5) > 10^38. - Martin Ehrenstein, Jun 24 2024

			1 = 1! + 0^2.
27 = 2! + 5^2.
729 = 6! + 3^2.
46656 = 7! + 204^2.

Cf. A000142, A000290, A000578, A167222.

isok(k) = my(c=k^3, i=1, p=1); while (1, if (issquare(c-p), return(1)); i++; p *=i; if (p> c, return(0)););
for (n=1, 50, if (isok(n), print1(n^3, ", "))); \\ Michel Marcus, Jun 06 2024

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A077121 Number of integer squares <= n^3.

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A167224 Table of primes of the form n^3 - k^2, 0<=k<=A077121(n).

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A261893 a(n) = (n+1)^3 - n^2.

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A373218 Cubes equal to the sum of a factorial number and a square.

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