A003059 -id:A003059 - cp's OEIS frontend

A295005 Numbers n such that the largest digit of n^2 is 5.

5, 15, 35, 39, 45, 50, 55, 65, 71, 105, 112, 115, 145, 150, 155, 185, 188, 205, 211, 229, 235, 335, 350, 365, 368, 388, 389, 390, 450, 461, 485, 495, 500, 501, 502, 505, 550, 579, 585, 595, 635, 650, 652, 665, 671, 710, 711, 715, 718, 729, 735, 745, 1005, 1015, 1050

Offset: 1

M. F. Hasler, Nov 12 2017

			39 is in this sequence because 39^2 = 1521 has 5 as largest digit.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A295015 (the corresponding squares), A277959 .. A277961 (same for digit 2 .. 4), A295006 .. A295009 (same for digit 6 .. 9).

Cf. A000290 (the squares).

Select[Sqrt[ #]&/@(FromDigits/@Select[Tuples[ Range[ 0,5],7],Max[#] == 5&]),IntegerQ] (* Harvey P. Dale, Sep 23 2021 *)

select( is_A295005(n)=n&&vecmax(digits(n^2))==5 , [0..999]) \\ The "n&&" avoids an error message for n=0.

def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    if max(str(k*k)) == "5": alst.append(k)
  return alst
print(aupto(1050)) # Michael S. Branicky, May 15 2021

a(n) = sqrt(A295015(n)), where sqrt = A000196 or A000194 or A003059.

A295009 Numbers k such that the largest digit of k^2 is 9.

Original entry on oeis.org

3, 7, 13, 14, 17, 23, 27, 30, 31, 33, 36, 37, 43, 44, 47, 53, 54, 57, 63, 64, 67, 70, 73, 77, 83, 86, 87, 89, 93, 95, 96, 97, 98, 99, 103, 107, 113, 114, 117, 118, 123, 127, 130, 133, 134, 136, 137, 138, 139, 140, 141, 143, 147, 148, 153, 157, 158, 161, 163, 164, 167, 170, 171

Offset: 1

M. F. Hasler, Nov 12 2017

			23 is in this sequence because 23^2 = 529 has 9 as largest digit.

Cf. A295019 (the corresponding squares), A277959 .. A277961 (same for digit 2 .. 4), A295005 .. A295008 (same for digit 5 .. 8).

Cf. A000290 (the squares).

select( is_A295009(n)=n&&vecmax(digits(n^2))==9 , [0..999]) \\ The "n&&" avoids an error message for n=0.

a(n) = sqrt(A295019(n)), where sqrt = A000196 or A000194 or A003059.

A017912 Powers of sqrt(2) rounded up.

Original entry on oeis.org

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

A302862 a(n) = [x^n] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 4, 8, 20, 57, 160, 422, 1076, 2780, 7449, 20462, 56348, 153909, 418268, 1139703, 3126068, 8618611, 23801146, 65708424, 181391905, 501296216, 1387834518, 3848187985, 10680579812, 29660831057, 82415406493, 229156296047, 637659848888, 1775648562970, 4947475298595

Offset: 0

Ilya Gutkovskiy, Apr 14 2018

nonn

a(n) = number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_n)^2 <= n.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to sums of squares

Cf. A000606, A003059, A010052, A224212, A224213, A287617, A302860, A302863.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/(2^n (1 - x)), {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, 0, n}]^n, {x, 0, n}], {n, 0, 30}]

A341400 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n.

Original entry on oeis.org

1, 6, 16, 26, 36, 57, 87, 107, 122, 157, 207, 247, 277, 322, 392, 452, 482, 537, 637, 717, 773, 863, 973, 1053, 1113, 1203, 1343, 1473, 1553, 1668, 1858, 1998, 2053, 2173, 2373, 2543, 2673, 2818, 3018, 3218, 3338, 3483, 3753, 3973, 4113, 4344, 4634, 4834, 4944, 5139, 5449

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A038671.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A038671, A055404, A055411, A175360, A224212, A224213, A302862, A341401, A341402.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 5)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 10 2021

nmax = 50; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^5/(32 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^5 / (32 * (1 - x)).

a(n^2) = A055404(n).

A341401 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n.

Original entry on oeis.org

1, 7, 22, 42, 63, 99, 160, 220, 265, 337, 457, 577, 672, 792, 978, 1178, 1319, 1469, 1739, 2039, 2255, 2507, 2882, 3242, 3513, 3819, 4269, 4769, 5159, 5555, 6181, 6841, 7246, 7666, 8401, 9181, 9763, 10363, 11188, 12108, 12828, 13434, 14394, 15534, 16359, 17211, 18477, 19677

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045848.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045848, A055405, A055412, A175361, A224212, A224213, A302862, A341400, A341402.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 6)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..47);  # Alois P. Heinz, Feb 10 2021

nmax = 47; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^6/(64 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^6 / (64 * (1 - x)).

a(n^2) = A055405(n).

A341402 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Original entry on oeis.org

1, 8, 29, 64, 106, 169, 281, 422, 548, 702, 961, 1276, 1556, 1864, 2326, 2893, 3390, 3852, 4545, 5455, 6253, 7002, 8080, 9361, 10453, 11496, 12903, 14618, 16194, 17643, 19589, 22011, 24027, 25714, 28143, 31188, 33792, 36137, 39203, 42920, 46294, 49108, 52580, 57165, 61365

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045849.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045849, A055406, A055413, A224212, A224213, A302862, A341400, A341401.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 7)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 10 2021

nmax = 44; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^7/(128 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^7 / (128 * (1 - x)).

a(n^2) = A055406(n).

A038760 a(n) = n - floor(sqrt(n)) * ceiling(sqrt(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, -1, 0, 1, 2, 0, -2, -1, 0, 1, 2, 3, 0, -3, -2, -1, 0, 1, 2, 3, 4, 0, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 0, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 0, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 0, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, -8, -7, -6, -5, -4

Offset: 0

Henry Bottomley, May 03 2000

			Sqrt(31) is between 5 and 6, and 31 - 6*5 = 1, so a(31)=1.

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

Cf. A053188.

a:= n-> n -(x-> floor(x)*ceil(x))(sqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 03 2015

f[n_]:=n-Floor[Sqrt[n]]*Ceiling[Sqrt[n]];Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2010 *)

a(n)=if(issquare(n),0,my(s=sqrtint(n));n-s^2-s) \\ Charles R Greathouse IV, Feb 07 2013

from math import isqrt
def A038760(n): return m-k if (m:=n-(k:=isqrt(n))**2) else 0 # Chai Wah Wu, Jul 28 2022

a(n) = n - A000196(n)*A003059(n) = n - A038759(n).

A170949 "Conway's Converger": a reordering of the integers (see Comments for definition).

Original entry on oeis.org

1, 3, 2, 4, 8, 6, 5, 7, 9, 15, 13, 11, 10, 12, 14, 16, 24, 22, 20, 18, 17, 19, 21, 23, 25, 35, 33, 31, 29, 27, 26, 28, 30, 32, 34, 36, 48, 46, 44, 42, 40, 38, 37, 39, 41, 43, 45, 47, 49, 63, 61, 59, 57, 55, 53, 51, 50, 52, 54, 56, 58, 60, 62, 64, 80, 78, 76, 74, 72

Offset: 1

N. J. A. Sloane, Feb 21 2010

The integers are written in blocks of lengths 1, 3, 5, 7, 9, ... . The first number in the block is moved to the center of the block, and then the numbers are written alternately to the left and the right. The block of length 2n-1 ends with n^2, which is not moved.
Let S = Sum_{i >= 1} s(i) be a not necessarily converging series and let T = Sum_{i >= 1} s(a(i)). Then if S converges so does T. On the other hand there are examples where T converges but S does not (for example S = 1 + 1 + 0 - 1 + 1/2 + 1/2 + 0 - 1/2 - 1/2 + 1/3 (3 times) + 0 - 1/3 (3 times) + 1/5 (5 times) + 0 - 1/5 (5 times) + ...). [Conway]
From Reinhard Zumkeller, Mar 08 2010: (Start)
a(n + 2*A003059(n)) = a(n) + 2*A003059(n) - 1;
a(A002522(n-1)) = A132411(n); a(A002061(n)) = A002522(n-1). (End)
The sum of the rows is n^3+(n+1)^3 [A005898] (1,9,35,91,189,...). - Vincenzo Librandi, Feb 22 2010

			                           1
                        3  2  4
                     8  6  5  7  9
                 15 13 11 10 12 14 16
              24 22 20 18 17 19 21 23 25
           35 33 31 29 27 26 28 30 32 34 36
        48 46 44 42 40 38 37 39 41 43 45 47 49
     63 61 59 57 55 53 51 50 52 54 56 58 60 62 64
  80 78 76 74 72 70 68 66 65 67 69 71 73 75 77 79 81

J. H. Conway, Personal communication, Feb 19 2010

R. Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers [From _Reinhard Zumkeller_, Mar 08 2010]

Cf. A170950, A009858.

Cf. A000290 (right diagonal), A132411 (left diagonal). - Michel Marcus, Aug 02 2018

a170949 n k = a170949_tabf !! (n-1) !! (k-1)
a170949_row n = a170949_tabf !! (n-1)
a170949_tabf = [1] : (map fst $ iterate f ([3,2,4], 3)) where
  f (xs@(x:_), i) = ([x + i + 2] ++ (map (+ i) xs) ++ [x + i + 3], i + 2)
a170949_list = concat a170949_tabf
-- Reinhard Zumkeller, Jan 31 2014

row[n_] := Join[ro = Range[n^2-1, (n-1)^2+1, -2], Reverse[ro]-1, {n^2}];
Array[row, 9] // Flatten (* Jean-François Alcover, Aug 02 2018 *)

A214078 a(n) = (ceiling (sqrt(n)))!.

Original entry on oeis.org

1, 1, 2, 2, 2, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 120, 120, 120, 120, 120, 120, 120, 120, 120, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 40320, 40320, 40320

Offset: 0

Mohammad K. Azarian, Dec 22 2012

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

Cf. A214078, A214079, A214080, A214081, A214083, A055228, A055226.

Cf. A000142, A003059.

PROG(y := [], n := 50, LOOP(IF(n = -1, RETURN y), y := ADJOIN(CEILING(SQRT(n))!, y), n := n - 1))

[Factorial(Ceiling (Sqrt(n))): n in [0..50]]; // Vincenzo Librandi, Feb 13 2013

Table[Ceiling[Sqrt[n]]!, {n, 0, 50}] (* T. D. Noe, Dec 23 2012 *)

a(n) = ceil(sqrt(n))!; \\ Altug Alkan, Jan 11 2016

from math import factorial, isqrt
def A214078(n): return factorial(1+isqrt(n-1)) if n else 1 # Chai Wah Wu, Jul 28 2022

a(n) = A000142(A003059(n)). - Michel Marcus, Jul 28 2022

Sum_{n>=0} 1/a(n) = e + 2. - Amiram Eldar, Aug 15 2022

cp's OEIS Frontend

A295005 Numbers n such that the largest digit of n^2 is 5.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A295009 Numbers k such that the largest digit of k^2 is 9.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A017912 Powers of sqrt(2) rounded up.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A302862 a(n) = [x^n] (1 + theta_3(x))^n/(2^n*(1 - x)), where theta_3() is the Jacobi theta function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A341400 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_5)^2 <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A341401 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_6)^2 <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A341402 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_7)^2 <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A038760 a(n) = n - floor(sqrt(n)) * ceiling(sqrt(n)).

Views

Author

Keywords

Examples