A003059 -id:A003059 - cp's OEIS frontend

A132983 a(n) = ceiling(n^(1/3) + n^(1/4)).

2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 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, 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

Offset: 1

Mohammad K. Azarian, Nov 18 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000196, A003059, A134914, A134915, A134916.

Table[Ceiling[Surd[n,3]+Surd[n,4]],{n,80}] (* Harvey P. Dale, May 07 2014 *)

for(n=1, 50, print1(ceil(n^(1/3) + n^(1/4)), ", ")) \\ G. C. Greubel, Sep 28 2017

A134919 Floor(n^(5/3)).

Original entry on oeis.org

1, 3, 6, 10, 14, 19, 25, 32, 38, 46, 54, 62, 71, 81, 91, 101, 112, 123, 135, 147, 159, 172, 186, 199, 213, 228, 243, 258, 273, 289, 305, 322, 339, 356, 374, 392, 410, 429, 448, 467, 487, 507, 527, 548, 569, 590, 612, 633, 656

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, A134918.

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

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

A262872 Expansion of (Sum_{i>=0} x^(i^2)) * (Sum_{j>=0} x^(j^3)) / (1-x).

Original entry on oeis.org

1, 3, 4, 4, 5, 6, 6, 6, 7, 9, 10, 10, 11, 11, 11, 11, 12, 14, 14, 14, 14, 14, 14, 14, 15, 16, 17, 18, 19, 19, 19, 20, 20, 21, 21, 21, 23, 24, 24, 24, 24, 24, 24, 25, 26, 26, 26, 26, 26, 27, 28, 28, 29, 29, 29, 29, 29, 30, 30, 30, 30, 30, 30, 31, 32, 33, 33, 33, 34, 34, 34

Offset: 0

Ran Pan, Oct 03 2015

nonn

a(n) is number of nonnegative integer solutions (x,y,z) such that x + y^2 + z^3 = n.

			a(4) = 5 because there are 5 solutions: (5,0,0), (4,1,0), (4,0,1), (3,1,1) and (1,2,0).

Cf. A003059, A262874.

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

A262874 Expansion of (Sum_{i>=0} x^(i^2)) * (Sum_{j>=0} x^(j^3)) * (Sum_{k>=0} x^(k^4)) / (1-x).

Original entry on oeis.org

1, 4, 7, 8, 9, 11, 12, 12, 13, 16, 19, 20, 21, 22, 22, 22, 24, 29, 32, 32, 33, 34, 34, 34, 36, 40, 43, 45, 48, 49, 49, 50, 52, 55, 56, 56, 58, 61, 62, 62, 63, 64, 65, 67, 70, 71, 71, 72, 72, 74, 76, 77, 80, 82, 82, 82, 82, 83, 84, 85, 86, 86, 86, 87, 89, 92, 94, 94, 96, 97, 97

Offset: 0

Ran Pan, Oct 03 2015

nonn

a(n) is number of nonnegative integer solutions (x,x,z,u) such that x + y^2 + z^3 + u^4 = n.

Cf. A003059, A262872.

G.f.: (Sum_{i>=0} x^(i^2)) * (Sum_{j>=0} x^(j^3)) * (Sum_{k>=0} x^(k^4)) / (1-x).

A342219 a(1) = 1, a(2) = 2; for n > 2, a(n) = the number of terms in the maximal length sum of previous consecutive terms that equals n.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 4, 3, 4, 5, 3, 5, 6, 5, 5, 6, 7, 3, 7, 8, 7, 6, 8, 9, 7, 8, 8, 9, 10, 8, 10, 11, 8, 10, 8, 11, 12, 10, 8, 11, 10, 12, 13, 8, 12, 11, 13, 14, 11, 13, 6, 14, 15, 13, 10, 14, 9, 15, 16, 11, 14, 14, 15, 15, 16, 17, 13, 17, 18, 10, 16, 9, 17, 15, 18, 19, 16, 15, 17, 15, 18, 13

Offset: 1

Scott R. Shannon, Mar 05 2021

nonn

The equivalent sequence for a minimal length sum is given by A003059.

			a(3) = 2 as the only way to sum previous consecutive terms to make 3 is 1 + 2 = 3, which contains two terms.
a(7) = 4 as the previous consecutive terms 1 + 2 + 2 + 2 = 7, which contains four terms. Note that 7 can also be made by consecutive terms 2 + 2 + 3 = 7, but the sequence is the maximal sum length.
a(10) = 5 as the previous consecutive terms 1 + 2 + 2 + 2 + 3 = 10, which contains five terms. Three other consecutive term sums also exist that sum to 10 but they contain fewer terms.

Scott R. Shannon, Scatterplot of the first 50000 terms.

Cf. A163169, A003059, A064816, A118139

A368941 a(n) = floor(3/2 + sqrt(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 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, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Eric W. Weisstein, Jan 10 2024

Burning number of the n-ladder (for n >= 1), n-Moebius ladder (for n >= 3), and n-prism (for n >= 3) graphs.

Eric Weisstein's World of Mathematics, Burning Number.
Eric Weisstein's World of Mathematics, Ladder Graph.
Eric Weisstein's World of Mathematics, Moebius Ladder.
Eric Weisstein's World of Mathematics, Prism Graph.

Sequence agrees with the known terms of A155934.

Cf. A000194, A003059.

Table[Floor[3/2 + Sqrt[n]], {n, 50}]
Floor[3/2 + Sqrt[Range[50]]]
CoefficientList[Series[(1 + QPochhammer[-x^2, x^4]  QPochhammer[x^8, x^8])/(1 - x), {x, 0, 50}], x]

a(n) = A000194(n) + 1. - Andrew Howroyd, Jan 10 2024

G.f.: x*(1 + QPochhammer(-x^2, x^4)*QPochhammer(x^8, x^8))/(1 - x).

Terms a(26) and beyond from Andrew Howroyd, Jan 10 2024

A123241 Number of squares < lesser of twin primes.

Original entry on oeis.org

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

Offset: 1

Giovanni Teofilatto, Oct 07 2006

Cf. A104103 number of squares < prime(n).

ltp=Select[Prime[Range[PrimePi[2500]]],PrimeQ[#+2]&];Ceiling[Sqrt[ltp]] (* James C. McMahon, Nov 17 2024 *)

a(n)=A003059(A001359(n)). - R. J. Mathar, Jun 18 2008

More terms from R. J. Mathar, Jun 18 2008

A130819 2k appears 2k-1 times.

Original entry on oeis.org

2, 4, 4, 4, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18

Offset: 1

Paul Curtz, Jul 17 2007

nonn

The run-length encoding of this sequence is A000027 as it has one 2, three 4's, five 6's, and so on. - Alexander Fraebel, Sep 10 2012

Equals twice A003059. - Michel Marcus, Sep 14 2020

Cf. A000027.

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2. - Amiram Eldar, Oct 01 2022

A132913 a(n) = ceiling(sqrt(n) + n^(1/3)).

Original entry on oeis.org

2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13

Offset: 1

Mohammad K. Azarian, Nov 18 2007

nonn

Cf. A000196, A003059, A134914, A134915, A134916.

Table[Ceiling[Sqrt[n]+Surd[n,3]],{n,70}] (* Harvey P. Dale, Jul 31 2018 *)

A132914 a(n) = floor(sqrt(n) + n^(1/3)).

Original entry on oeis.org

2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12

Offset: 1

Mohammad K. Azarian, Nov 18 2007

Cf. A000196, A003059, A134914, A134915, A134916.

Table[Floor[Sqrt[n] + n^(1/3)], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2024 *)

cp's OEIS Frontend

A132983 a(n) = ceiling(n^(1/3) + n^(1/4)).

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A134919 Floor(n^(5/3)).

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A262872 Expansion of (Sum_{i>=0} x^(i^2)) * (Sum_{j>=0} x^(j^3)) / (1-x).

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A262874 Expansion of (Sum_{i>=0} x^(i^2)) * (Sum_{j>=0} x^(j^3)) * (Sum_{k>=0} x^(k^4)) / (1-x).

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A342219 a(1) = 1, a(2) = 2; for n > 2, a(n) = the number of terms in the maximal length sum of previous consecutive terms that equals n.

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A368941 a(n) = floor(3/2 + sqrt(n)).

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Mathematica

Formula

Extensions

A123241 Number of squares < lesser of twin primes.

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Extensions

A130819 2*k appears 2*k-1 times.

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A132913 a(n) = ceiling(sqrt(n) + n^(1/3)).

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Mathematica

A132914 a(n) = floor(sqrt(n) + n^(1/3)).

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A130819 2k appears 2k-1 times.