A000148 -id:A000148

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A000158

Number of partitions into non-integral powers.
(Formerly M1830 N0726)

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

COMMENTS

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

REFERENCES

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).

LINKS

Table of n, a(n) for n=3..43.

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]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Sean A. Irvine, Oct 11 2009

STATUS

approved

A000327

Number of partitions into non-integral powers.
(Formerly M3819 N1563)

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,2

COMMENTS

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

REFERENCES

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).

LINKS

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]

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A000148, A000158, A000160.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A000135

Number of partitions into non-integral powers.
(Formerly M1595 N0622)

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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_1<x_2<x_3<x_4<... - R. J. Mathar, Jul 03 2009

REFERENCES

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).

LINKS

Table of n, a(n) for n=1..38.

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

EXAMPLE

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

CROSSREFS

Cf. A000148, A000158, A000160.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

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

STATUS

approved

A259447

Triangle read by rows arising from enumeration of partitions into non-integral powers.

+10
0

1, 2, 1, 5, 2, 1, 8, 7, 2, 1, 11, 15, 8, 2, 1, 14, 28, 19, 8, 2, 1, 18, 45, 41, 21, 8, 2, 1, 22, 70, 78, 48, 22, 8, 2, 1, 27, 100, 134, 99, 52, 22, 8, 2, 1, 31, 138, 218, 186, 111, 53, 22, 8, 2, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..55.

B. K. Agarwala, F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. See Table 1.

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]

EXAMPLE

Triangle begins:

2,1,

5,2,1,

8,7,2,1,

11,15,8,2,1,

14,28,19,8,2,1,

18,45,41,21,8,2,1,

22,70,78,48,22,8,2,1,

27,100,134,99,52,22,8,2,1,

31,138,218,186,111,53,22,8,2,1,

...

CROSSREFS

Columns include A000093, A000148, A000158, A000169.

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jun 27 2015

STATUS

approved

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