A075847 -id:A075847 - cp's OEIS frontend

A077108 Duplicate of A075847.

0, 0, 3, 1, 8, 17, 9, 22, 0, 17, 36, 57, 19, 44, 71, 9, 40, 73, 108, 18, 57, 98, 141, 17, 64

Offset: 0

dead

A077116 n^3 - A065733(n).

0, 0, 4, 2, 0, 4, 20, 19, 28, 0, 39, 35, 47, 81, 40, 11, 0, 13, 56, 135, 79, 45, 39, 67, 135, 0, 152, 83, 48, 53, 104, 207, 7, 216, 100, 26, 0, 28, 116, 270, 496, 277, 104, 546, 503, 524, 615, 139, 368, 0, 391, 155, 732, 652, 648, 726, 55, 293, 631, 170, 704

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) = 0 for n = m^2. - Zak Seidov, May 11 2007

It has been asked whether some primes do not occur in this sequence. It seems indeed that primes 3, 5, 17, 23, 29, 31, 37, 41, 43, 59, 61,... do not occur, primes 2, 7, 11, 13, 19, 47, 53, 67, 79, 83,... do. For further investigations, see A087285 = the range of this sequence, and also the related sequences A229618 = range of A181138, and A165288. - M. F. Hasler, Sep 26 2013 and Oct 05 2013

			A065733(10) = 961 = 31^2 is the largest square less than or equal to 10^3 = 1000, therefore a(10) = 1000 - 961 = 39.

Cf. A000578, A070929, A077118, A077119, A075847.

A077116 := proc(n)
    A053186(n^3) ;
end proc: # R. J. Mathar, Jul 12 2016

Table[c = n^3; c - Floor[Sqrt[c]]^2, {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 02 2011 *)

A077116(n)=n^3-sqrtint(n^3)^2 \\ - M. F. Hasler, Sep 26 2013

a(n) = A154333(n) unless n is a square or, equivalently, a(n)=0. - M. F. Hasler, Oct 05 2013

a(n) = A053186(n^3). - R. J. Mathar, Jul 12 2016

A070923 a(n) is the smallest value >= 0 of the form x^3 - n^2.

Original entry on oeis.org

0, 0, 4, 18, 11, 2, 28, 15, 0, 44, 25, 4, 72, 47, 20, 118, 87, 54, 19, 151, 112, 71, 28, 200, 153, 104, 53, 0, 216, 159, 100, 39, 307, 242, 175, 106, 35, 359, 284, 207, 128, 47, 433, 348, 261, 172, 81, 535, 440, 343, 244, 143, 40, 566, 459, 350, 239, 126, 11, 615, 496

Offset: 0

Benoit Cloitre, May 20 2002

a(n) = 0 if n is a cube (i.e., n is in A000578(k)).

a(n) = A181138(n) if n is not a cube. - Zak Seidov, Mar 26 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000290, A068527, A070929, A075847, A077110, A077111, A181138.

a(n) = ceiling(n^(2/3))^3 - n^2 = A077107(n)-n^2.

a(0)=0 prepended by Alois P. Heinz, Mar 07 2022

A077110 Nearest integer cube to n^2.

Original entry on oeis.org

0, 1, 1, 8, 8, 27, 27, 64, 64, 64, 125, 125, 125, 125, 216, 216, 216, 343, 343, 343, 343, 512, 512, 512, 512, 729, 729, 729, 729, 729, 1000, 1000, 1000, 1000, 1000, 1331, 1331, 1331, 1331, 1331, 1728, 1728, 1728, 1728, 1728, 2197, 2197, 2197

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(10) = 125, as 125 = 5^3 is the nearest cube to 100 = 10^2.

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

Cf. A000578, A000290, A075847, A070923, A077106, A077107, A077111, A077118.

Cf. A002760 (Squares and cubes). - Zak Seidov, Oct 08 2015

nic[n_]:=Module[{n2=n^2,s3,c1,c2},s3=Surd[n2,3];c1=Floor[s3]^3;c2= Ceiling[ s3]^3;If[n2-c1Harvey P. Dale, Jul 05 2015 *)

from sympy import integer_nthroot
def A077110(n):
    n2 = n**2
    a = integer_nthroot(n2,3)[0]
    a2, a3 = a**3, (a+1)**3
    return a3 if a3+a2-2*n2 < 0 else a2 # Chai Wah Wu, Sep 24 2021

a(n) = if A075847(n) < A070923(n) then A077106(n) else A077107(n).

A077106 Largest integer cube <= n^2.

Original entry on oeis.org

0, 1, 1, 8, 8, 8, 27, 27, 64, 64, 64, 64, 125, 125, 125, 216, 216, 216, 216, 343, 343, 343, 343, 512, 512, 512, 512, 729, 729, 729, 729, 729, 1000, 1000, 1000, 1000, 1000, 1331, 1331, 1331, 1331, 1331, 1728, 1728, 1728, 1728, 1728, 2197, 2197

Offset: 0

Reinhard Zumkeller, Oct 29 2002

			a(20) = 343, as 343 = 7^3 is the largest cube <= 400 = 20^2.

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

Cf. A000578, A000290, A065733, A075847, A077107, A077110, A077113.

a[n_] := Floor[Surd[n^2, 3]]^3; Array[a, 100, 0] (* Amiram Eldar, Apr 05 2025 *)

a(n) + A075847(n) = n^2.

a(n) = A100196(n)^3. - Amiram Eldar, Apr 06 2025

A077111 a(n) = A077110(n) - n^2.

Original entry on oeis.org

0, 0, -3, -1, -8, 2, -9, 15, 0, -17, 25, 4, -19, -44, 20, -9, -40, 54, 19, -18, -57, 71, 28, -17, -64, 104, 53, 0, -55, -112, 100, 39, -24, -89, -156, 106, 35, -38, -113, -190, 128, 47, -36, -121, -208, 172, 81, -12, -107, -204, 244, 143, 40, -65, -172

Offset: 0

Reinhard Zumkeller, Oct 29 2002

sign

a(n)=0 iff n = m^(6*k).

			A077110(20)=343=7^3 is the nearest cube to 20^2=400, therefore a(20)=343-400=-57.

Cf. A000290, A077110, A077119.

a(n) = if A075847(n) < A070923(n) then -A075847(n) else A070923(n).

A176582 n^2+Largest cube, (Largest cube <= n^2).

Original entry on oeis.org

0, 2, 5, 17, 24, 33, 63, 76, 128, 145, 164, 185, 269, 294, 321, 441, 472, 505, 540, 704, 743, 784, 827, 1041, 1088, 1137, 1188, 1458, 1513, 1570, 1629, 1690, 2024, 2089, 2156, 2225, 2296, 2700, 2775, 2852, 2931, 3012, 3492, 3577, 3664, 3753, 3844, 4406

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

9+8=17, 16+8=24, 25+8=33, 36+27=63,..

Cf. A065733, A075847, A077116, A176580, A176581.

r[n_]:=n^2;f[n_]:=r[n]+Floor[r[n]^(1/3)]^3;Table[f[n],{n,0,5!}]

A176583 n^2+Smallest cube, (Smallest cube >= n^2).

Original entry on oeis.org

0, 2, 12, 36, 43, 52, 100, 113, 128, 206, 225, 246, 360, 385, 412, 568, 599, 632, 667, 873, 912, 953, 996, 1258, 1305, 1354, 1405, 1458, 1784, 1841, 1900, 1961, 2355, 2420, 2487, 2556, 2627, 3097, 3172, 3249, 3328, 3409, 3961, 4046, 4133, 4222, 4313, 4953

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

4+8=12, 9+27=36, 16+27=43,..

Cf. A065733, A075847, A077116, A176580, A176581, A176582

r[n_]:=n^2;f[n_]:=r[n]+Ceiling[r[n]^(1/3)]^3;Table[f[n],{n,0,5!}]
Table[n^2+Ceiling[Surd[n^2,3]]^3,{n,0,50}] (* Harvey P. Dale, Jun 04 2016 *)

A176584 Primes of the form p = k^3 + (largest square <= k^3).

Original entry on oeis.org

2, 194057, 601067, 1607173, 3283993, 3630257, 3721363, 4918999, 5255713, 5848307, 6749569, 9999907, 17992913, 20432729, 23393483, 34746203, 44845993, 73843813, 84277273, 107165033, 109735649, 120469409, 125138873, 130898927

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

Consider k = 4 * x^2 + 1, where the largest square <= k^3 is (8 * x^3 + 3 * x)^2. Bunyakowsky's conjecture implies there are infinitely many primes of the form (4 * x^2 + 1)^3 + (8 * x^3 + 3 * x)^2, and so that the sequence is infinite. The first term of this form is 93389778901 = a(165), corresponding to x = 30. - Robert Israel, Jan 10 2024

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

Primes in A176580.

Cf. A065733, A075847, A077116, A176581, A176582, A176583.

g:= n -> n + floor(sqrt(n))^2:
select(isprime,map(g,[seq(i^3,i=1..1000)])); # Robert Israel, Jan 10 2024

r[n_]:=n^3;f[n_]:=r[n]+Floor[Sqrt[r[n]]]^2;Select[Table[f[n],{n,0,6!}],PrimeQ[ # ]&]

select(isprime, vector(1000, n, n^3+sqrtint(n^3)^2)) \\ Michel Marcus, Jan 10 2024

A176585 Primes of the form n^3+Smallest square, (Smallest square >= n^3).

Original entry on oeis.org

2, 17, 269, 281233, 1770217, 1826609, 2520841, 3907529, 7595017, 8665471, 9828089, 11280377, 12259063, 17235221, 27654961, 54008809, 70583033, 75196799, 85018949, 87240233, 106316057, 111499057, 168061561, 176255669, 201105409

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 21 2010

nonn

8+9=17, 5^3+12^2=269,..

Cf. A065733, A075847, A077116, A176580, A176581, A176582, A176583, A176584

r[n_]:=n^3;f[n_]:=r[n]+Ceiling[Sqrt[r[n]]]^2;Select[Table[f[n],{n,0,6!}],PrimeQ[ # ]&]
ssn3[n_]:=n^3+(Ceiling[Sqrt[n^3]])^2; Select[Array[ssn3,500],PrimeQ] (* Harvey P. Dale, Jun 23 2017 *)

cp's OEIS Frontend

A077108 Duplicate of A075847.

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A077116 n^3 - A065733(n).

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A070923 a(n) is the smallest value >= 0 of the form x^3 - n^2.

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A077110 Nearest integer cube to n^2.

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A077106 Largest integer cube <= n^2.

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A077111 a(n) = A077110(n) - n^2.

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A176582 n^2+Largest cube, (Largest cube <= n^2).

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A176583 n^2+Smallest cube, (Smallest cube >= n^2).

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A176584 Primes of the form p = k^3 + (largest square <= k^3).

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