A329202 -id:A329202 - cp's OEIS frontend

A017912 Powers of sqrt(2) rounded up.

1, 2, 2, 3, 4, 6, 8, 12, 16, 23, 32, 46, 64, 91, 128, 182, 256, 363, 512, 725, 1024, 1449, 2048, 2897, 4096, 5793, 8192, 11586, 16384, 23171, 32768, 46341, 65536, 92682, 131072, 185364, 262144, 370728, 524288, 741456, 1048576, 1482911, 2097152, 2965821, 4194304

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A017910, A329202.

[Ceiling(Sqrt(2^n)): n in [0..40]]; // Vincenzo Librandi, Nov 20 2011

Ceiling[(Sqrt[2])^Range[0,40]] (* Vincenzo Librandi, Nov 20 2011 *)

from math import isqrt
def A017912(n): return isqrt(1<>1) # Chai Wah Wu, Jul 26 2022

a(n) = A003059(A000079(n)). - Jason Kimberley, Oct 28 2016

a(n) = A017910(n)+1 if n is odd. a(n) = A017910(n) = 2^(n/2) if n is even. - Chai Wah Wu, Jul 26 2022

A328911 Irregular triangle read by rows: T(n,k) = number of solutions to Erdös's Last Equation x_1...x_n = n(x_1+...+x_n), 0 < x_1 <= ... <= x_n, having k+1 components x_i > 1, 1 <= k <= 2log_2(n).

Original entry on oeis.org

2, 0, 3, 3, 0, 4, 3, 1, 0, 4, 4, 0, 0, 6, 7, 4, 0, 0, 6, 5, 3, 0, 0, 5, 7, 4, 2, 1, 0, 8, 13, 5, 1, 0, 0, 9, 12, 3, 1, 0, 0, 6, 6, 3, 0, 0, 0, 8, 13, 9, 3, 0, 0, 0, 8, 7, 1, 0, 0, 0, 0, 6, 15, 6, 2, 1, 0, 0, 12, 16, 12, 3, 0, 0, 0, 12, 15, 11, 4, 2, 1, 0, 0, 6, 8, 2, 2, 0, 0, 0, 0

Offset: 2

M. F. Hasler, Nov 07 2019

For n = 1 the equation is trivially solved by any integer, therefore we only consider n >= 2.
If any x_k = 0, then all x_i must be zero, so (0, ..., 0) would be the only additional solution in nonnegative integers. This solution is not considered here.
A vector (1, ..., 1, x_n) can never be a solution for n > 1. The number of components different from 1 must be k+1 >= 2 <=> k >= 1.
It can be shown that no solution can have 2^k > n^2, cf. the Shiu paper. Therefore row lengths are floor(2 log_2(n)) = (2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, ...) = A329202(n), n >= 2.
Row sums yield the total number of nontrivial solutions A328910(n), see there for more information.
T(n,k) is equal to |C_k(n)| in the Shiu paper, but some values given in the table on top of p. 803 are erroneous (pers. comm. from the author).

			The table starts:
   n : T(n,k), 1 <= k <= 2*log_2(n)
   2 :   2   0
   3 :   3   3   0
   4 :   4   3   1   0
   5 :   4   4   0   0
   6 :   6   7   4   0   0
   7 :   6   5   3   0   0
   8 :   5   7   4   2   1   0
   9 :   8  13   5   1   0   0
  10 :   9  12   3   1   0   0
  11 :   6   6   3   0   0   0
  12 :   8  13   9   3   0   0   0
  13 :   8   7   1   0   0   0   0
  14 :   6  15   6   2   1   0   0
  15 :  12  16  12   3   0   0   0
For n = 2 variables, we have the equation x1*x2 = 2*(x1 + x2) with positive integer solutions (3,6) and (4,4): Both have k+1 = 2 components > 1, i.e., k = 1.
For n = 3, we have T(3,1) = 3 solutions with k+1 = 2 components > 1, {(1, 4, 15), (1, 5, 9), (1, 6, 7)}, and T(3,2) = 3 with k+1 = 3 components > 1, {(2, 2, 12), (2, 3, 5), (3,3,3)}.
For n = 4 we have the 8 solutions (1, 1, 5, 28), (1, 1, 6, 16), (1, 1, 7, 12), (1, 1, 8, 10), (1, 2, 3, 12), (1, 2, 4, 7), (1, 3, 4, 4) and (2, 2, 2, 6). Four of them have k+1 = 2 components > 1, i.e., k = 1, whence T(4,1) = 4. Three have k+1 = 3 <=> k = 2, so T(4,2) = 3. One has k+1 = 4, so T(4,3) = 1.
For n = 5, the solutions are, omitting initial components x_i = 1: {(6, 45), (7, 25), (9, 15), (10, 13), (2, 3, 35), (2, 5, 9), (3, 3, 10), (3, 5, 5)}. Therefore T(5,1..4) = (4, 4, 0, 0).
For n = 6, the solutions are (omitting x_i = 1): {(7, 66), (8, 36), (9, 26), (10, 21), (11, 18), (12, 16), (2, 4, 27), (2, 5, 15), (2, 6, 11), (2, 7, 9), (3, 3, 18), (3, 4, 10), (3, 6, 6), (2, 2, 2, 24), (2, 2, 3, 9), (2, 2, 4, 6), (2, 3, 3, 5)}. Therefore T(6,1..5) = (6, 7, 4, 0, 0).
For n = 9, the 27 solutions are (omitting '1's): {(10, 153), (11, 81), (12, 57), (13, 45), (15, 33), (17, 27), (18, 25), (21, 21), (2, 5, 117), (2, 6, 42), (2, 7, 27), (2, 9, 17), (2, 12, 12), (3, 4, 39), (3, 5, 21), (3, 6, 15), (3, 7, 12), (3, 9, 9), (4, 4, 18), (5, 5, 9), (6, 6, 6), (2, 2, 3, 36), (2, 2, 6, 9), (2, 3, 3, 13), (3, 3, 3, 7), (3, 3, 4, 5), (2, 3, 3, 3, 3)}. Therefore T(9,1..6) = (8, 13, 5, 1, 0, 0).

David A. Corneth, Table of n, a(n) for n = 2..14645 (first 899 rows flattened)
Peter Shiu, On Erdös's Last Equation, Amer. Math. Monthly, 126 (2019), 802-808.

Cf. A328910, A329202.

A328911(n,k,show=1)={if( k0 && (n*vecsum(x)+t)%d==0 && (n*vecsum(x)+t)\d >= x[k] && s++&& show&& printf("%d,",concat(x,(n*vecsum(x)+t)\d)),1);s)}

A329193 a(n) = floor(log_2(n^3)) = floor(3 log_2(n)).

Original entry on oeis.org

0, 3, 4, 6, 6, 7, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18

Offset: 1

M. F. Hasler, Nov 07 2019

nonn

3*A000523(n) <= A000523(n^3) = a(n) <= A004257(n^3) <= A029837(n^3) <= 3*A029837(n) with equality for powers of 2 (A000079) and asymptotic equivalence as n -> oo.

Cf. A000578 (n^3), A000523 (floor log_2), A004257 (round log_2), A029837 (ceiling log_2), A329202 (log_2(n^2)).

apply( A329193(n)=exponent(n^3), [1..99]) \\ exponent(.) = logint(.,2) = log(.)\log(2)

A329194 a(n) = floor(log_3(n^2)) = floor(2 log_3(n)).

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 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, 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, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

M. F. Hasler, Nov 07 2019

nonn

Cf. A000290 (n^2), A062153 (log_3), A329202 (log_2(n^2)), A329193 (log_2(n^3)).

Table[Floor[Log[3,n^2]],{n,120}] (* Harvey P. Dale, May 04 2025 *)

apply( A329194(n)=logint(n^2,3), [1..99])

2*A062153(n) <= a(n) = floor(log_3(n^2)) = A062153(A000290(n)).

This A329194 = A062153 o A000290.

A225668 a(n) = floor(4*log_2(n)).

Original entry on oeis.org

0, 4, 6, 8, 9, 10, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 24

Offset: 1

Jonathan Vos Post, May 11 2013

Arises in analysis of "when to clean your room".

			a(3) = floor(4*log_2(3)) = floor(6.33985000) = 6.
a(8) = floor(4*log_2(8)) = floor(4*3) = 12.

Kimball Martin and Krishnan Shankar, How often should you clean your room?, arXiv:1305.1984 [math.CO], 2013-2014.

Cf. A000583 (n^4), A000523 (floor log_2), A004257 (round log_2), A029837 (ceiling log_2).

Cf. A329202 (log_2(n^2)), A329193 (log_2(n^3)).

A225668 := proc(n)
    4*log[2](n) ;
    floor(%) ;
end proc: # R. J. Mathar, May 12 2013

Table[Floor[4*Log[2, n]], {n, 1, 64}] (* Jean-François Alcover, Nov 30 2017 *)

a(n) = 4*log(n)\log(2); \\ Michel Marcus, Nov 30 2017

apply( A225668(n)=exponent(n^4), [1..99]) \\ M. F. Hasler, Nov 07 2019

a(n) = floor(4*log(n)/log(2)).

a(n) = floor(log_2(n^4)) = A000523(A000583(n)), i.e., this A225668 = A000523 o A000583. - M. F. Hasler, Nov 07 2019

Better definition from M. F. Hasler, Nov 07 2019

A329195 a(n) = floor(log_5(n^2)) = floor(2 log_5(n)).

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Nov 07 2019

nonn

Cf. A000290 (n^2), A062153 (log_3), A329202 (log_2(n^2)), A329193 (log_2(n^3)), A329194 (log_3(n^2)).

Table[Floor[2Log[5,n]],{n,100}] (* Harvey P. Dale, Dec 14 2021 *)

apply( A329195(n)=logint(n^2,5), [1..99])

cp's OEIS Frontend

A017912 Powers of sqrt(2) rounded up.

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A328911 Irregular triangle read by rows: T(n,k) = number of solutions to Erdös's Last Equation x_1*...*x_n = n*(x_1+...+x_n), 0 < x_1 <= ... <= x_n, having k+1 components x_i > 1, 1 <= k <= 2*log_2(n).

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A329193 a(n) = floor(log_2(n^3)) = floor(3 log_2(n)).

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A329194 a(n) = floor(log_3(n^2)) = floor(2 log_3(n)).

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A225668 a(n) = floor(4*log_2(n)).

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A329195 a(n) = floor(log_5(n^2)) = floor(2 log_5(n)).

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PARI

A328911 Irregular triangle read by rows: T(n,k) = number of solutions to Erdös's Last Equation x_1...x_n = n(x_1+...+x_n), 0 < x_1 <= ... <= x_n, having k+1 components x_i > 1, 1 <= k <= 2log_2(n).