A000093 -id:A000093 - cp's OEIS frontend

A074704 a(n) = floor(n^(3/2)) - n*floor(n^(1/2)).

0, 0, 2, 0, 1, 2, 4, 6, 0, 1, 3, 5, 7, 10, 13, 0, 2, 4, 6, 9, 12, 15, 18, 21, 0, 2, 5, 8, 11, 14, 17, 21, 24, 28, 32, 0, 3, 6, 9, 12, 16, 20, 23, 27, 31, 35, 40, 44, 0, 3, 7, 10, 14, 18, 22, 27, 31, 35, 40, 44, 49, 54, 59, 0, 4, 8, 12, 16, 21, 25, 30, 34, 39, 44, 49, 54, 59, 64, 70

Offset: 1

Reinhard Zumkeller, Sep 03 2002

a(n) is the coefficient of 1/n in the base n representation of sqrt(n); namely, the digit after the 'decimal' point. - Jason Kimberley, Feb 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A000196 [the digit before the 'decimal' point in the base n representation of sqrt(n)], A000093.

a(n) = A000093(n) - n * A000196(n).

A000263 Number of partitions into non-integral powers.

Original entry on oeis.org

3, 14, 39, 91, 173, 307, 502, 779, 1150, 1651, 2280, 3090, 4090, 5313, 6787, 8564, 10643, 13103, 15948, 19235, 23000, 27316, 32174, 37677, 43849, 50758, 58427, 66978, 76373, 86765, 98171, 110662, 124310, 139202, 155339, 172885

Offset: 3

N. J. A. Sloane

nonn

a(n) counts the solutions to the inequality x_1^(1/2)+x_2^(1/2)<=n for any two distinct integers 1<=x_1R. J. Mathar, Jul 03 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. K. Agarwala, F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

A000263 := proc(n) local a,x1,x2 ; a := 0 ; for x1 from 1 to n^2 do x2 := (n-x1^(1/2))^2 ; if floor(x2) >= x1+1 then a := a+floor(x2-x1) ; fi; od: a ; end: seq(A000263(n),n=3..80) ; # R. J. Mathar, Sep 29 2009

A000263[n_] := Module[{a, x1, x2 }, a = 0; For[x1 = 1, x1 <= n^2, x1++, x2 = (n-x1^(1/2))^2; If[Floor[x2] >= x1+1, a = a+Floor[x2-x1]]]; a]; Table[ A000263[n], {n, 3, 80}] (* Jean-François Alcover, Feb 06 2016, after R. J. Mathar *)

More terms from R. J. Mathar, Sep 29 2009

A000339 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 5, 18, 45, 100, 185, 323, 522, 804, 1180, 1687, 2322, 3139, 4146, 5377, 6859, 8645, 10733, 13203, 16058, 19356, 23132, 27460, 32330, 37846, 44031, 50954, 58637, 67203, 76613, 87021, 98443, 110951, 124616, 139526, 155681, 173246, 192243

Offset: 2

N. J. A. Sloane

nonn

a(n) counts the solutions to the inequality x_1^(1/2)+x_2^(1/2)<=n for any two integers 1<=x_1<=x_2. - R. J. Mathar, Jul 03 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Delbert L. Johnson, Table of n, a(n) for n = 2..7032
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

A000339 := proc(n) local a,x1,x2 ; a := 0 ; for x1 from 1 to n^2 do x2 := (n-x1^(1/2))^2 ; if floor(x2) >= x1 then a := a+floor(x2-x1+1) ; fi; od: a ; end: for n from 2 to 80 do printf("%d,\n",A000339(n)) ; od: # R. J. Mathar, Sep 29 2009

A000339[n_] := Module[{a, x1, x2}, a = 0; For[x1 = 1 , x1 <= n^2 , x1++, x2 = (n-x1^(1/2))^2; If[Floor[x2] >= x1, a = a+Floor[x2-x1+1]]]; a]; Reap[ For[n = 2, n <= 80, n++, Print[an = A000339[n]]; Sow[an]]][[2, 1]] (* Jean-François Alcover, Feb 07 2016, after R. J. Mathar *)

More terms from R. J. Mathar, Sep 29 2009

A000397 Number of partitions into non-integral powers.

Original entry on oeis.org

6, 32, 109, 288, 654, 1337, 2506, 4414, 7379, 11822, 18273, 27356, 39938, 56974, 79607, 109267, 147523, 196295, 257715, 334407, 429086, 545034, 685917, 855886, 1059360, 1301776, 1588321, 1925620, 2320544, 2780468, 3314007, 3930001, 4638319, 5449943, 6376505, 7430471, 8625369, 9976540, 11498855, 13210238, 15128487, 17272896, 19664754, 22326319, 25280987, 28554486, 32173404, 36166409, 40563607, 45397395, 50701682, 56512012, 62866699, 69805531, 77370606, 85607286, 94560129, 104280410, 114819255, 126229853, 138570284, 151899428, 166278945, 181775849, 198456941, 216394746, 235661505, 256338017, 278503009, 302242623, 327644632, 354799834, 383805368, 414759214, 447764499, 482931051

Offset: 5

N. J. A. Sloane

nonn

a(n) counts the solutions to the inequality x_1^(1/2)+x_2^(1/2)+x_3^(1/2)<=n for any three distinct integers 1<=x_1R. J. Mathar, Jul 03 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. K. Agarwala, F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

A000397 := proc(n) local a,x1,x2,x3 ; a := 0 ; for x1 from 1 to n^2 do for x2 from x1+1 to floor( (n-x1^(1/2))^2 ) do x3 := (n-x1^(1/2)-x2^(1/2))^2 ; if floor(x3) >= x2+1 then a := a+floor(x3-x2) ; fi; od: od: a ; end: for n from 5 do printf("%d,\n",A000397(n)) ; od: # R. J. Mathar, Sep 29 2009

A000397[n_] := Module[{a, x1, x2, x3}, a = 0; For[x1 = 1, x1 <= n^2, x1++, For[x2 = x1+1, x2 <= Floor[(n-x1^(1/2))^2], x2++, x3 = (n-x1^(1/2) - x2^(1/2))^2 ; If[Floor[x3] >= x2+1, a = a + Floor[x3-x2]]]]; a]; Reap[ For[n = 5, n <= 40, n++, Print[an = A000397[n]; Sow[an]]]][[2, 1]] (* Jean-François Alcover, Feb 08 2016, after R. J. Mathar *)

More terms from R. J. Mathar, Sep 29 2009

More terms from Sean A. Irvine, Nov 14 2010

A238170 Integer part of square root of A001017: a(n) = floor(n^(9/2)).

Original entry on oeis.org

0, 1, 22, 140, 512, 1397, 3174, 6352, 11585, 19683, 31622, 48558, 71831, 102978, 143739, 196069, 262144, 344365, 445375, 568056, 715541, 891223, 1098758, 1342070, 1625363, 1953125, 2330129, 2761448, 3252453, 3808824, 4436552, 5141947, 5931641, 6812597, 7792110

Offset: 0

Philippe Deléham, Feb 21 2014

nonn

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

Integer part of square root of n^k: A000196 (k=1), A000093 (k=3), A155013 (k=5), A155014 (k=7), this sequence (k=9), A155015 (k=11), A155016 (k=13), A155018 (k=15), A155019 (k=17).

[Floor(n^(9/2)): n in [0..40]]; // Vincenzo Librandi, Feb 23 2014

Table[Floor[n^(9/2)], {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

a(n) = floor(n^(9/2)); \\ Joerg Arndt, Feb 23 2014

from math import isqrt
def A238170(n): return isqrt(n**9) # Chai Wah Wu, Jan 27 2023

a(n) = floor(n^(9/2)).
a(n) = A000196(A001017(n)).
a(n) = floor(n^4*sqrt(n)).

A283295 Integers n such that floor(n^(3/2)) is squarefree.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 23, 32, 41, 43, 44, 45, 46, 47, 50, 52, 53, 55, 56, 57, 59, 69, 71, 72, 73, 75, 76, 80, 82, 84, 86, 87, 89, 90, 94, 97, 98, 99, 101, 102, 103, 106, 107, 108, 109, 110, 111, 112, 113, 114, 116, 117, 118, 119, 122, 125

Offset: 1

Michel Marcus, Mar 04 2017

nonn

Cao & Zhai prove that this sequence is infinite.

The asymptotic density of this sequence is 6/Pi^2 (A059956) (Cao and Zhai, 1998). - Amiram Eldar, Feb 17 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zhai, The distribution of square-free numbers of the form [n^c], Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2 , p. 287-299.

Cf. A000093, A005117, A059956, A283296.

Select[Range[200],SquareFreeQ[Floor[#^(3/2)]]&] (* Harvey P. Dale, Aug 25 2017 *)

PARI
```
isok(n) = issquarefree(sqrtint(n^3));
```

A283296 Integers n such that floor(n^(3/2)) is cubefree.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 65, 67, 69, 70, 71, 72, 73, 74, 75, 76, 80, 82, 84, 86, 87, 88, 89, 90

Offset: 1

Michel Marcus, Mar 04 2017

nonn

Zang & Li prove that this sequence is infinite.

The asymptotic density of this sequence is 1/zeta(3) (A088453) (Zang and Li, 2017). - Amiram Eldar, Feb 17 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Min Zhang and Jinjiang Li, Distribution of cube-free numbers with form [n^c], Frontiers of Mathematics in China, Vol. 12 (2017), pp. 1515-1525; arXiv preprint, arXiv:1702.00165 [math.NT], 2017.

Cf. A000093, A004709, A088453, A283295.

Select[Range[90], Max[FactorInteger[Floor[#^(3/2)]][[All, 2]]] < 3&] (* Jean-François Alcover, Feb 23 2019 *)

isok(n) = if (n < 4, 1, vecmax(factor(sqrtint(n^3))[,2]) < 3);

A352718 a(n) = floor(n^(3/2)) - floor(n^(1/2))^3.

Original entry on oeis.org

0, 0, 1, 4, 0, 3, 6, 10, 14, 0, 4, 9, 14, 19, 25, 31, 0, 6, 12, 18, 25, 32, 39, 46, 53, 0, 7, 15, 23, 31, 39, 47, 56, 64, 73, 82, 0, 9, 18, 27, 36, 46, 56, 65, 75, 85, 95, 106, 116, 0, 10, 21, 31, 42, 53, 64, 76, 87, 98, 110, 121, 133, 145, 157, 0, 12, 24

Offset: 0

Clark Kimberling, Jun 21 2022

a(n) = 0 if and only if n is a square.

Cf. A000093, A352718.

Table[Floor[n^(3/2)] - Floor[n^(1/2)]^3, {n, 0, 120}]

a(n) = sqrtint(n^3) - sqrtint(n)^3; \\ Michel Marcus, Jun 24 2022

A000135 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 2, 6, 13, 24, 42, 73, 125, 204, 324, 511, 801, 1228, 1856, 2780, 4135, 6084, 8873, 12847, 18481, 26416, 37473, 52871, 74216, 103596, 143841, 198839, 273654, 374987, 511735, 695559, 941932, 1271139, 1709474, 2291195, 3061385, 4078152, 5416322

Offset: 1

N. J. A. Sloane

nonn

a(n) counts the solutions to the inequality sum_{i=1,2,..} x_i^(2/3)<=n for any number of distinct integers 1<=x_1R. J. Mathar, Jul 03 2009

			For n=3, the 6 solutions are (i) 1^(2/3)<=3. (ii) 1^(2/3)+2^(2/3)<=3. (iii) 2^(2/3)<=3. (iv) 3^(2/3)<=3. (v) 4^(2/3)<=3. (vi) 5^(2/3)<=3. - _R. J. Mathar_, Jul 03 2009

B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. K. Agarwala, F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
Sean A. Irvine, Tentative values of first 55 terms

Cf. A000148, A000158, A000160.

8 more terms from R. J. Mathar, Jul 03 2009

20 more terms from Sean A. Irvine, Sep 28 2009

A000298 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 4, 12, 30, 70, 159, 339, 706, 1436, 2853, 5551, 10622, 19975, 37043, 67811, 122561, 219090, 387578, 678977, 1178769, 2029115, 3465056, 5872648, 9882301, 16517284, 27430358, 45275673, 74297072, 121245153, 196810381, 317850809, 510830685, 817139589, 1301251186, 2063204707, 3257690903, 5123047561

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of solutions to the inequality sum_{i=1,2,..} x_i^(1/2)<=n for unknowns 1<=x_1R. J. Mathar, Jul 03 2009

			The 12 solutions for n=3 are 1^(1/2)<=3, 1^(1/2)+2^(1/2)<=3, 1^(1/2)+3^(1/2)<=3, 1^(1/2)+4^(1/2)<=3, 2^(1/2)<=3, 3^(1/2)<=3,...,8^(1/2)<=3 and 9^(1/2)<=3. - _R. J. Mathar_, Jul 03 2009

B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. K. Agarwala, F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

3 more terms from R. J. Mathar, Jul 03 2009

More terms from Sean A. Irvine, Nov 11 2010

cp's OEIS Frontend

A074704 a(n) = floor(n^(3/2)) - n*floor(n^(1/2)).

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A000263 Number of partitions into non-integral powers.

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Maple

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A000339 Number of partitions into non-integral powers.

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Maple

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A000397 Number of partitions into non-integral powers.

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Maple

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A238170 Integer part of square root of A001017: a(n) = floor(n^(9/2)).

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Magma

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PARI

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A283295 Integers n such that floor(n^(3/2)) is squarefree.

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Mathematica

PARI

A283296 Integers n such that floor(n^(3/2)) is cubefree.

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Mathematica

PARI

A352718 a(n) = floor(n^(3/2)) - floor(n^(1/2))^3.

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