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, 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]
-
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 *)
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, 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]
-
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
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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 *)
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
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
A259447
Triangle read by rows arising from enumeration of partitions into non-integral powers.
Original entry on oeis.org
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
Offset: 1
Triangle begins:
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,
...
Showing 1-4 of 4 results.
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