A000093 -id:A000093 - cp's OEIS frontend

A155019 Integer part of square root of n^17 = A010805(n).

0, 1, 362, 11363, 131072, 873464, 4114202, 15252229, 47453132, 129140163, 316227766, 710947978, 1489500287, 2941158941, 5521897022, 9926032708, 17179869184, 28761784747, 46753733110, 74029634996, 114486680447

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 19 2009

nonn

Alois P. Heinz, 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), A155015 (k=11), A155016 (k=13), A155018 (k=15), this sequence (k=17).

[Floor(Sqrt(n^17)): n in [0..30]]; // G. C. Greubel, Dec 30 2017

a={};Do[AppendTo[a,IntegerPart[(n^17)^(1/2)]],{n,0,5!}];a
Table[Floor[Sqrt[n^17]], {n,0,30}] (* G. C. Greubel, Dec 30 2017 *)

for(n=0,30, print1(floor(sqrt(n^17)), ", ")) \\ G. C. Greubel, Dec 30 2017

Offset corrected by Alois P. Heinz, Sep 27 2014

A000148 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 2, 7, 15, 28, 45, 70, 100, 138, 183, 242, 310, 388, 481, 583, 701, 838, 984, 1152, 1337, 1535, 1757, 2001, 2262, 2545, 2855, 3183, 3540, 3926, 4335, 4770, 5233, 5728, 6248, 6801, 7388, 8005, 8658, 9345, 10064, 10824, 11620, 12452, 13324, 14236

Offset: 2

N. J. A. Sloane

nonn

a(n) is the number of solutions to the inequality x_1^(2/3) + x_2^(2/3) <= n where 1 <= x_1 <= x_2 are any two integers. If the number of terms in the sum is not restricted to 2, we get A000234. - 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).

Seth A. Troisi, Table of n, a(n) for n = 2..1000
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]

Cf. A000158, A000160, A000327.

A000148[n_] := Sum[Min[xi, Floor[(n - xi^(2/3))^(3/2)]], {xi, 1, Floor[n^(3/2)]}];
Table[A000148[n], {n, 2, 100}] (* Seth A. Troisi, May 25 2022 *)

More terms from Sean A. Irvine, Oct 08 2009

A000336 a(n) = a(n-1)a(n-2)a(n-3)*a(n-4); for n < 5, a(n) = n.

Original entry on oeis.org

1, 2, 3, 4, 24, 576, 165888, 9172942848, 21035720123168587776, 18437563379178327736384102280592359424, 590180110002114158896983994712576414865667267958188575935810179040280576

Offset: 1

N. J. A. Sloane

nonn

The next term has 139 digits. - Harvey P. Dale, Jan 21 2019

T. D. Noe, Table of n, a(n) for n = 1..15
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]

Cf. A000301, A000308, A001631, A003586, A251656.

A000336 := proc(n) option remember; if n <=4 then n else A000336(n-1)*A000336(n-2)*A000336(n-3)*A000336(n-4); fi; end;

t = {1, 2, 3, 4}; Do[AppendTo[t, t[[-1]]*t[[-2]]*t[[-3]]*t[[-4]]], {n, 5, 15}] (* T. D. Noe, Jun 19 2012 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,a b c d}; NestList[nxt,{1,2,3,4},10][[All,1]] (* Harvey P. Dale, Jan 21 2019 *)

a(n,a=[24,1,2,3,4])={for(n=6,n,a[n%5+1]=a[(n-1)%5+1]^2\a[n%5+1]);a[n%5+1]} \\ M. F. Hasler, Apr 22 2018

first(n) = n = max(n, 5); my(res = vector(n)); for(i=1, 4, res[i] = i); res[5]=24; for(i = 6, n, res[i] = res[i-1]^2 / res[i - 5]); res \\ David A. Corneth, Apr 22 2018

a(n) = 2^A251656(n) * 3^A001631(n-1). - Vaclav Kotesovec, Feb 02 2016

a(n) = a(n-1)^2 / a(n-5), for n > 5. - M. F. Hasler, Apr 22 2018

A185549 a(n) = ceiling(n^(3/2)); complement of A185550.

Original entry on oeis.org

0, 1, 3, 6, 8, 12, 15, 19, 23, 27, 32, 37, 42, 47, 53, 59, 64, 71, 77, 83, 90, 97, 104, 111, 118, 125, 133, 141, 149, 157, 165, 173, 182, 190, 199, 208, 216, 226, 235, 244, 253, 263, 273, 282, 292, 302, 312, 323, 333, 343, 354, 365, 375, 386, 397, 408, 420, 431, 442, 454, 465, 477, 489

Offset: 0

Clark Kimberling, Jan 30 2011

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000093, A077107, A185550.

Cf. A144968 (first differences).

a185549 = ceiling . (** (3 / 2)) . fromIntegral
-- Reinhard Zumkeller, Jul 24 2015

f[n_]=Ceiling[n^(3/2)];
t1=Table[f[n],{n,1,90}];t1  (* A185549 *)
t2=Complement[Range[150], Table[f[n],{n,1,80}]];t2  (* A185550 *)
Ceiling[Sqrt[Range[0,70]^3]] (* Harvey P. Dale, Jul 04 2023 *)

for(n=0,50, print1(ceil(n^(3/2)), ", ")) \\ G. C. Greubel, Jul 08 2017

a(n) = ceiling(n^(3/2)).

a(n) = sqrt(A077107(n)). - Amiram Eldar, May 17 2025

Initial a(0)=0 prepended and offset adjusted by Reinhard Zumkeller, Jul 24 2015

A355159 Numbers k such that (fractional part of k^(3/2)) < 1/2.

Original entry on oeis.org

0, 1, 3, 4, 5, 9, 11, 14, 15, 16, 17, 18, 20, 21, 22, 23, 25, 27, 28, 29, 30, 32, 34, 35, 36, 37, 38, 42, 47, 49, 51, 56, 57, 59, 61, 62, 63, 64, 65, 66, 67, 69, 71, 79, 81, 83, 87, 91, 92, 94, 97, 98, 99, 100, 101, 102, 103, 106, 108, 111, 112, 113, 114, 115

Offset: 0

Clark Kimberling, Jun 22 2022

For each nonnegative integer K there is a greatest nonnegative integer h such that h/K <= sqrt(K); a(n) is the n-th number h such that h/K is closer to sqrt(K) than (h+1)/K is.

Cf. A000093, A355160 (complement).

Select[-1 + Range[300], N[FractionalPart[#^(3/2)]] < 1/2 &]  (* A355159 *)
Select[-1 + Range[300], N[FractionalPart[#^(3/2)]] > 1/2 &]  (* A355160 *)

isok(k) = frac(k^(3/2)) < 1/2; \\ Michel Marcus, Jul 11 2022

from math import isqrt
from itertools import count, islice
def A355159_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:int(((r:=n**3)-(m:=isqrt(r))*(m+1))<<2<=1),count(max(startvalue,0)))
A355159_list = list(islice(A355159_gen(),30)) # Chai Wah Wu, Aug 03 2022

A000158 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 2, 8, 19, 41, 78, 134, 218, 339, 506, 730, 1023, 1397, 1884, 2477, 3218, 4118, 5192, 6486, 8010, 9795, 11888, 14302, 17066, 20256, 23889, 27999, 32637, 37863, 43695, 50218, 57495, 65545, 74431, 84257, 95030, 106840, 119799, 133941, 149311, 166071

Offset: 3

N. J. A. Sloane

nonn

a(n) counts the solutions to the inequality x_1^(2/3)+x_2^(2/3)+x_3^(2/3)<=n for any three integers 1<=x_1<=x_2<=x_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]

Cf. A000148, A000160, A000234. - R. J. Mathar, Jul 03 2009

More terms from Sean A. Irvine, Oct 11 2009

A000160 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 2, 8, 21, 48, 99, 186, 330, 556, 895, 1397, 2107, 3097, 4459, 6264, 8644, 11760, 15742, 20790, 27128, 34993, 44664, 56473, 70784, 87995, 108564, 132970, 161828, 195686, 235274, 281349, 334682, 396202, 466849, 547712, 639935, 744716, 863443

Offset: 4

N. J. A. Sloane

nonn

a(n) counts the solutions to the inequality x_1^(2/3)+x_2^(2/3)+x_3^(2/3)+x_4^(2/3)<=n for any four integers 1<=x_1<=x_2<=x_3<=x_4. - 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]

More terms from Sean A. Irvine, Oct 11 2009

A355160 Numbers k such that (fractional part of k^(3/2)) > 1/2.

Original entry on oeis.org

2, 6, 7, 8, 10, 12, 13, 19, 24, 26, 31, 33, 39, 40, 41, 43, 44, 45, 46, 48, 50, 52, 53, 54, 55, 58, 60, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 89, 90, 93, 95, 96, 104, 105, 107, 109, 110, 117, 118, 120, 122, 124, 125, 132, 133, 135, 137

Offset: 1

Clark Kimberling, Jun 23 2022

For each positive integer K there is a greatest integer h such that h/K < sqrt(K); a(n) is the n-th number h such that (h+1)/K is closer to sqrt(K) than h/K is.

Cf. A000093, A355159 (complement).

Select[Range[300], N[FractionalPart[#^(3/2)]] < 1/2 &]  (* A355159 *)
Select[Range[300], N[FractionalPart[#^(3/2)]] > 1/2 &]  (* A355160 *)

isok(k) = frac(k^(3/2)) > 1/2; \\ Michel Marcus, Jul 11 2022

from math import isqrt
from itertools import count, islice
def A355160_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:int(((r:=n**3)-(m:=isqrt(r))*(m+1))<<2>1),count(max(startvalue,0)))
A355160_list = list(islice(A355160_gen(),30)) # Chai Wah Wu, Aug 03 2022

A000234 Partitions into non-integral powers (see Comments for precise definition).

Original entry on oeis.org

1, 3, 8, 18, 37, 72, 136, 251, 445, 770, 1312, 2202, 3632, 5908, 9501, 15111, 23781, 37083, 57293, 87813, 133530, 201574, 302265, 450317, 666743, 981488, 1437003, 2092976, 3033253, 4375104, 6282026, 8981046, 12786327, 18131492, 25612628

Offset: 1

N. J. A. Sloane

nonn

This sequence gives the number of solutions to the inequality Sum_{i=1,2,...} xi^(2/3) <= n with the constraint that 1 <= x1 <= x2 <= x3 <= ... is a list of at least 1 and no more than n integers. - R. J. Mathar, Oct 19 2007

			a(3)=8 counts 5 partitions with 1 term, explicitly { 1^(2/3), 2^(2/3), 3^(2/3), 4^(2/3), 5^(2/3) }, 2 partitions into sums of 2 terms { 1^(2/3) + 1^(2/3), 1^(2/3) + 2^(2/3) } and one partition into a sum of three terms { 1^(2/3) + 1^(2/3) + 1^(2/3) }.

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 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]

fs:=n->floor(simplify(n)): a:=proc(i, m, k) options remember: local s,l,j,m2: if(k=1) then RETURN(1) else s:=0: l:=fs(m^(3/2)): for j from 1 to min(l,i) do m2:=m-j^(2/3): if(fs(m2)>=1) then s:=s+a(j, m2, k-1) fi: s:=s+1 od: RETURN(s) fi: end: seq(a(fs(n^(3/2)), n, n),n=1..19); # Herman Jamke (hermanjamke(AT)fastmail.fm), May 03 2008

fs[n_] := Floor[Simplify[n]]; a[i_, m_, k_] := a[i, m, k] = Module[{s, l, j, m2}, If[k == 1, Return[1], s = 0; l = fs[m^(3/2)]; For[j = 1, j <= Min[l, i], j++, m2 = m - j^(2/3); If[fs[m2] >= 1, s = s + a[j, m2, k-1] ]; s = s+1]; Return[s]]]; A000234 = Table[an = a[fs[n^(3/2)], n, n]; Print["a(", n, ") = ", an]; an, {n, 1, 19}] (* Jean-François Alcover, Feb 06 2016, after Herman Jamke *)

More terms from R. J. Mathar, Oct 19 2007
One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 03 2008
a(20)-a(35) from Jon E. Schoenfield, Jan 17 2009

A000327 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 5, 12, 23, 39, 62, 91, 127, 171, 228, 294, 370, 461, 561, 677, 811, 955, 1121, 1303, 1499, 1719, 1960, 2218, 2499, 2806, 3131, 3485, 3868, 4274, 4706, 5166, 5658, 6175, 6725, 7309, 7923, 8572, 9256, 9972, 10728, 11521, 12349, 13218, 14126, 15072

Offset: 3

N. J. A. Sloane

nonn

a(n) counts the solutions to the inequality x_1^(2/3) + x_2^(2/3) <= n for any two distinct 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).

Seth A. Troisi, Table of n, a(n) for n = 3..1000
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]

Cf. A000148, A000158, A000160.

A000327 := proc(n) local a,x1,x2 ; a := 0 ; for x1 from 1 to floor(n^(3/2)) do x2 := (n-x1^(2/3))^(3/2) ; if floor(x2) >= x1+1 then a := a+floor(x2-x1) ; fi; od: a ; end: seq(A000327(n),n=3..80) ; # R. J. Mathar, Sep 29 2009

A000327[n_] := Module[{a, x1, x2 }, a = 0; For[x1 = 1, x1 <= Floor[ n^(3/2)], x1++, x2 = (n - x1^(2/3))^(3/2); If[Floor[x2] >= x1+1, a = a + Floor[x2 - x1]]]; a ]; Table[A000327[n], {n, 3, 80}] (* Jean-François Alcover, Feb 07 2016, after R. J. Mathar *)
A000327[n_] := Sum[Min[x1 - 1, Floor[(n - x1^(2/3))^(3/2)]], {x1, 2, Floor[n^(3/2)]}];
Table[A000327[n], {n, 3, 80}] (* Seth A. Troisi, May 25 2022 *)

a(n) = A000148(n) - floor((n/2)^(3/2)). - Seth A. Troisi, May 25 2022

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

cp's OEIS Frontend

A155019 Integer part of square root of n^17 = A010805(n).

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A000148 Number of partitions into non-integral powers.

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A000336 a(n) = a(n-1)*a(n-2)*a(n-3)*a(n-4); for n < 5, a(n) = n.

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A185549 a(n) = ceiling(n^(3/2)); complement of A185550.

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A355159 Numbers k such that (fractional part of k^(3/2)) < 1/2.

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A000158 Number of partitions into non-integral powers.

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A000160 Number of partitions into non-integral powers.

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A355160 Numbers k such that (fractional part of k^(3/2)) > 1/2.

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A000336 a(n) = a(n-1)a(n-2)a(n-3)*a(n-4); for n < 5, a(n) = n.