A028391 -id:A028391 - cp's OEIS frontend

A135678 Floor(n^(4/3)+n).

2, 4, 7, 10, 13, 16, 20, 24, 27, 31, 35, 39, 43, 47, 51, 56, 60, 65, 69, 74, 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 144, 149, 154, 160, 165, 171, 176, 182, 187, 193, 199, 205, 210, 216, 222, 228, 234, 240

Offset: 1

Mohammad K. Azarian, Dec 01 2007

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

Cf. A135660; A135661; A135662; A135663; A135664; A135665; A028391; A135668; A135671; A135672; A135673 ; A135674, A135675; A135676; A135677.

[Floor(n^(4/3)+n): n in [1..100]]; // Vincenzo Librandi, Feb 18 2013

Table[Floor[n^(4/3) + n], {n, 100}] (* Vincenzo Librandi, Feb 18 2013 *)

A139179 Number of non-fourth-powers <= n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 0

Jonathan Vos Post, Jun 06 2008

Cf. A028391, A064524.

a[n_] := n - Floor[n^(1/4)]; a /@ Range[0, 72] (* Giovanni Resta, Jun 21 2016 *)

from sympy import integer_nthroot
def A139179(n): return n-integer_nthroot(n,4)[0] # Chai Wah Wu, Jun 18 2024

a(n) = n - floor(n^(1/4)).

Offset corrected by Giovanni Resta, Jun 21 2016

A317369 a(0) = 0, a(1) = 1; for n >= 2, a(n) = freq(a(n-s(n)),n) where s = A000196 and freq(i, j) is the number of times i appears in the terms a(0) .. a(j-1).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 6, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 7, 7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 13, 13, 13, 13, 13, 13, 6, 6, 6, 6, 6, 6, 6, 20, 20, 20, 20, 20, 20, 20, 7, 7, 7, 7, 7, 7, 7, 7, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

Altug Alkan, Jul 26 2018

nonn

Altug Alkan, A line graph of a(n) for n <= 50000
Altug Alkan, A line graph of a(n) - A000196(n) + 1

Cf. A000196, A028391, A317223, A317359.

Nest[Append[#1, Count[#1, #1[[-Floor@ Sqrt@ #2]] ]] & @@ {#, Length@ #} &, {0, 1}, 92] (* Michael De Vlieger, Jul 27 2018 *)

A338621 Triangle read by rows: A(n, k) is the number of partitions of n with "aft" value k (see comments).

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 3, 2, 2, 4, 3, 2, 2, 4, 5, 2, 2, 2, 4, 6, 7, 1, 2, 2, 4, 6, 9, 6, 1, 2, 2, 4, 6, 10, 11, 7, 2, 2, 4, 6, 10, 13, 14, 5, 2, 2, 4, 6, 10, 14, 19, 15, 5, 2, 2, 4, 6, 10, 14, 21, 22, 17, 3, 2, 2, 4, 6, 10, 14, 22, 27, 29, 17, 2, 2, 2, 4, 6, 10, 14, 22, 29, 36, 33, 17

Offset: 0

Joshua Swanson, Nov 04 2020

The "aft" of an integer partition is the number of cells minus the larger of the number of parts or the largest part. For example, aft(4, 2, 2) = 8-4 = 4 = aft(3, 3, 1, 1).
Columns stabilize to twice the partition numbers: A(n, k) = 2p(n) = A139582(n) if n > 2k.
Row sums are partition numbers A000041.
Maximum value of k in row n is n - ceiling(sqrt(n)) = (n-1) - floor(sqrt(n-1)) = A028391(n-1).

			A(6, 2) = 4 since there are four partitions with 6 cells and aft 2, namely (4, 2), (2, 2, 1, 1), (4, 1, 1), (3, 1, 1, 1).
Triangle starts:
  1;
  1;
  2;
  2, 1;
  2, 2, 1;
  2, 2, 3;
  2, 2, 4, 3;
  2, 2, 4, 5,  2;
  2, 2, 4, 6,  7,  1;
  2, 2, 4, 6,  9,  6,  1;
  2, 2, 4, 6, 10, 11,  7;
  2, 2, 4, 6, 10, 13, 14,  5;
  2, 2, 4, 6, 10, 14, 19, 15,  5;
  2, 2, 4, 6, 10, 14, 21, 22, 17,  3;
  2, 2, 4, 6, 10, 14, 22, 27, 29, 17,  2;
  2, 2, 4, 6, 10, 14, 22, 29, 36, 33, 17,  1;
  2, 2, 4, 6, 10, 14, 22, 30, 41, 45, 39, 15,  1;
  2, 2, 4, 6, 10, 14, 22, 30, 43, 52, 57, 41, 14;
  2, 2, 4, 6, 10, 14, 22, 30, 44, 57, 69, 67, 47, 11;
  2, 2, 4, 6, 10, 14, 22, 30, 44, 59, 76, 85, 81, 46, 9; ...

S. C. Billey, M. Konvalinka, and J. P. Swanson, Asymptotic normality of the major index on standard tableaux, Adv. in Appl. Math. 113 (2020).

S. C. Billey, M. Konvalinka, and J. P. Swanson, Asymptotic normality of the major index on standard tableaux, arXiv:1905.00975 [math.CO], 2019.
FindStat - Combinatorial Statistic Finder, The aft of an integer partition

Cf. A000041, A139582.

CoefficientList[
SeriesCoefficient[
  1 + Sum[If[r == 0, 1, 2] q^(r + 1) Sum[
      q^(2 s) t^s QBinomial[2 s + r, s, q t], {s, 0, 30}], {r, 0,
     30}], {q, 0, 20}], t]

Row(n)={if(n==0, [1], my(v=vector(n)); forpart(p=n, v[1+n-max(#p, p[#p])]++); Vecrev(Polrev(v)))}
{ for(n=1, 15, print(Row(n))) } \\ Andrew Howroyd, Nov 04 2020

G.f.: Sum_{lambda} t^aft(lambda) * q^|lambda| = 1 + Sum_{r >= 0} c_r * q^(r+1) * Sum_{s >= 0} q^(2*s) * t^s * [2*s + r, s]_(q*t) where c_0 = 1, c_r = 2 for r >= 1, and [a, b]_q is a Gaussian binomial coefficient (see A022166).

cp's OEIS Frontend

A135678 Floor(n^(4/3)+n).

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A139179 Number of non-fourth-powers <= n.

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A317369 a(0) = 0, a(1) = 1; for n >= 2, a(n) = freq(a(n-s(n)),n) where s = A000196 and freq(i, j) is the number of times i appears in the terms a(0) .. a(j-1).

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A338621 Triangle read by rows: A(n, k) is the number of partitions of n with "aft" value k (see comments).

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