A000093 -id:A000093 - cp's OEIS frontend

A000345 Number of partitions into non-integral powers.

1, 5, 22, 71, 186, 427, 888, 1704, 3053, 5203, 8476, 13318, 20265, 29946, 43254, 61171, 84832, 115713, 155382, 205779, 269065, 347906, 445001, 563685, 707637, 881042, 1088339, 1335019, 1626233, 1968701, 2369320, 2835467, 3375820, 3999234, 4715586, 5535965, 6472005, 7536195, 8742102, 10105163, 11640190, 13365254, 15298155, 17458190, 19866739, 22546131, 25519743, 28813410, 32453730, 36469433, 40890672, 45749944, 51081147, 56919908, 63304577, 70275008, 77873381, 86145156, 95134772, 104893757, 115473250, 126926418, 139311512, 152687434, 167115830, 182663928, 199398527, 217392226, 236717247, 257454630, 279683011, 303488723, 328959602, 356186407

Offset: 3

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 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]

More terms from Sean A. Irvine, Nov 14 2010

A000347 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 5, 24, 84, 251, 653, 1543, 3341, 6763, 12879, 23446, 40883, 68757, 111976, 177358, 273926, 413784, 612430, 889959, 1271709, 1789841, 2483779, 3402623, 4605954, 6166614, 8171174, 10724604, 13950011, 17994136, 23029141, 29255902, 36908235, 46257694, 57616522, 71344257, 87853381, 107612397

Offset: 4

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) + x_4^(1/2) <= 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, Nov 14 2010

A000348 Number of ways to pair up {1^2, 2^2, ..., (2n)^2 } so sum of each pair is prime.

Original entry on oeis.org

1, 1, 2, 4, 12, 9, 72, 160, 428, 2434, 3011, 10337, 126962, 264182, 783550, 5004266, 34340141, 176302123, 1188146567, 4457147441, 7845512385, 132253267889, 1004345333251, 3865703506342, 40719018858150, 213982561376958, 1266218151414286, 10976172953868304, 59767467676582641, 512279001476451101, 6189067229056357433

Offset: 1

S. J. Greenfield (greenfie(AT)math.rutgers.edu)

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]
L. E. Greenfield and S. J. Greenfield, Some Problems of Combinatorial Number Theory Related to Bertrand's Postulate, J. Integer Sequences, 1998, #98.1.2.

Cf. A000341.

a[n_] := Permanent[Table[Boole[PrimeQ[(2*i)^2 + (2*j - 1)^2]], {i, 1, n}, {j, 1, n}]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 22}] (* Jean-François Alcover, Jan 06 2016, after T. D. Noe *)

permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;nc=0;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;nc+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p)
for(n=1,24,a=matrix(n,n,i,j,isprime((2*i)^2+(2*j-1)^2));print1(permRWNb(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007

a(n) = permanent(m), where the n X n matrix m is defined by m(i,j) = 1 or 0, depending on whether (2i)^2+(2j-1)^2 is prime or composite, respectively. - T. D. Noe, Feb 10 2007

a(11)-a(16) from David W. Wilson
a(17)-a(22) from T. D. Noe, Feb 10 2007
a(23)-a(24) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007
More terms from Sean A. Irvine, Nov 14 2010

A035936 Number of squares in (n^3, (n+1)^3 ].

Original entry on oeis.org

1, 1, 3, 3, 3, 3, 4, 4, 5, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 9, 10, 10, 10, 11, 10, 11, 10, 11, 10, 11, 11, 11, 12, 11, 11, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 12, 13, 12, 13, 13, 13, 13, 13, 13, 14, 13, 14, 13, 14

Offset: 0

Erich Friedman

There are never exactly two squares between two consecutive cubes. - Vladimir Pletser, Jan 12 2021

			a(3)=3 since 3^3 < 6^2, 7^2, 8^2 <= 4^3.

Vladimir Pletser, Table of n, a(n) for n = 0..10000

Cf. A000093, A000290 (squares), A000578 (cubes).

for n from 0 to 10000 do print(n, floor((n+1)^(3/2))-floor(n^(3/2))) end do; # Vladimir Pletser, Jan 11 2021

With[{sqs=Range[800]^2},Table[Count[sqs,?(#>n^3&& #<=(n+1)^3&)],{n,0,85}]]  (* _Harvey P. Dale, Apr 12 2011 *)

a(n) = A000093(n+1)-A000093(n) (first differences of A000093). - Henry Bottomley, Aug 31 2000

A060903 a(n) = floor(6nsqrt(n)/Pi^2).

Original entry on oeis.org

0, 0, 1, 3, 4, 6, 8, 11, 13, 16, 19, 22, 25, 28, 31, 35, 38, 42, 46, 50, 54, 58, 62, 67, 71, 75, 80, 85, 90, 94, 99, 104, 110, 115, 120, 125, 131, 136, 142, 148, 153, 159, 165, 171, 177, 183, 189, 195, 202, 208, 214, 221, 227, 234, 241, 247, 254, 261, 268, 275, 282

Offset: 0

Henry Bottomley, May 05 2001

nonn

Conjecture: the sum of the divisors of n is less than a(n) for n exceeding 12. - Robert G. Wilson v, May 14 2014

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000093, A000203, A000796, A058208.

f[n_] := Floor[6 n^(3/2)/Pi^2]; Array[f, 61, 0] (* Robert G. Wilson v, May 14 2014 *)

{ default(realprecision, 100); t=Pi^2/6; for (n=0, 1000, write("b060903.txt", n, " ", n*sqrt(n)\t); ) } \\ Harry J. Smith, Jul 14 2009

a(n) = A000203(n) + A058208(n).

a(n) = floor(6*n^(3/2)/Pi^2).

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 5, 4, 1, 1, 2, 5, 8, 9, 5, 1, 1, 2, 6, 11, 16, 15, 8, 1, 1, 2, 7, 14, 25, 32, 27, 11, 1, 1, 3, 8, 18, 36, 55, 64, 46, 16, 1, 1, 3, 9, 22, 49, 88, 125, 128, 81, 22, 1, 1, 3, 10, 27, 64, 129, 216, 279, 256, 140, 32, 1, 1, 3, 11, 31, 81, 181, 343, 529, 625, 512, 243, 45, 1

Offset: 0

Kival Ngaokrajang, Feb 28 2015

			See table in the links.

G. C. Greubel, Table of n, a(n) for the first 100 antidaigonals, flattened
Kival Ngaokrajang, Example of table T(n,k), n = 0..12, k = 1..10

Rows(1..10): A000012, A000196, A000027, A000093, A000290, A155013, A000578, A155014, A000583, A238170.
Columns(1..10): A000012, A017910, A017913, A000079, A017919, A017922, A017925, A017928, A000244, A017934.
Diagonals: A190343, A066642, A075264.

T[n_, k_] := Floor[Sqrt[k^n]]; Table[T[k, n + 1 - k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Dec 30 2017 *)

{for(i=1,20,for(n=0,i-1,a=floor(sqrt((i-n)^n));print1(a,", ")))}

T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Terms a(81) onward added by G. C. Greubel, Dec 30 2017

A000333 Number of partitions into non-integral powers.

Original entry on oeis.org

1, 5, 15, 40, 98, 237, 534, 1185, 2554, 5391, 11117, 22556, 44858, 88000, 170107, 324547, 611755, 1140382, 2103554, 3842826, 6955918, 12483075, 22220002, 39248230, 68819781, 119839422, 207304370, 356356801, 608901907, 1034452712, 1747764522, 2937370605, 4911675955, 8173032301

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of solutions to the inequality sum_{i=1,2,3...} x_i^(1/2)<=n under the constraint that x_i are integers where 1<=x_1<=x_2<=x_3<=x_4<=... [From R. J. Mathar, Jul 03 2009]

			a(n=3)=15 counts the solutions 1^(1/2)<=3, 1^(1/2)+1^(1/2)<=3, 1^(1/2)+1^(1/2)+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, 2^(1/2)+2^(1/2)<=3, 3^(1/2)<=3, 4^(1/2)<=3,.., 8^(1/2)<=3 and 9^(1/2)<=3. [From _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]

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

More terms from Sean A. Irvine, Nov 14 2010

A063038 a(n) = floor(n*sqrt(n)) - d(n), where d(n) is the number of divisors function.

Original entry on oeis.org

0, 0, 3, 5, 9, 10, 16, 18, 24, 27, 34, 35, 44, 48, 54, 59, 68, 70, 80, 83, 92, 99, 108, 109, 122, 128, 136, 142, 154, 156, 170, 175, 185, 194, 203, 207, 223, 230, 239, 244, 260, 264, 279, 285, 295, 307, 320, 322, 340, 347, 360, 368, 383, 388, 403, 411, 426, 437

Offset: 1

Jason Earls, Aug 03 2001

Cf. A000005, A000093, A055682.

Table[Floor[n*Sqrt[n]] - DivisorSigma[0, n], {n, 50}] (* Wesley Ivan Hurt, Jun 09 2014 *)

j=[]; for(n=1,100,j=concat(j,floor(n*sqrt(n))-numdiv(n))); j

a(n) = A000093(n) - A000005(n). - Michel Marcus, Apr 13 2024

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

N. J. A. Sloane, Jun 27 2015

			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,
...

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]

Columns include A000093, A000148, A000158, A000169.

A259448 Triangle read by rows arising from enumeration of partitions into non-integral powers.

Original entry on oeis.org

1, 4, 1, 9, 5, 1, 16, 18, 5, 1, 25, 45, 22, 5, 1, 36, 100, 71, 24, 5, 1, 49, 185, 186, 84, 24, 5, 1, 64, 323, 427, 251, 90, 24, 5, 1, 81, 522, 888, 653, 288, 92, 24, 5, 1, 100, 804, 1704, 1543, 811, 306, 93, 24, 5, 1

Offset: 1

N. J. A. Sloane, Jun 27 2015

			Triangle begins:
1,
4,1,
9,5,1,
16,18,5,1,
25,45,22,5,1,
36,100,71,24,5,1,
49,185,186,84,24,5,1,
64,323,427,251,90,24,5,1,
81,522,888,653,288,92,24,5,1,
100,804,1704,1543,811,306,93,24,5,1,
...

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 2.
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]

Columns include A000339, A000345, A000347.

cp's OEIS Frontend

A000345 Number of partitions into non-integral powers.

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A000347 Number of partitions into non-integral powers.

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A000348 Number of ways to pair up {1^2, 2^2, ..., (2n)^2 } so sum of each pair is prime.

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Mathematica

PARI

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A035936 Number of squares in (n^3, (n+1)^3 ].

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Maple

Mathematica

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A060903 a(n) = floor(6*n*sqrt(n)/Pi^2).

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Mathematica

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A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

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A000333 Number of partitions into non-integral powers.

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A063038 a(n) = floor(n*sqrt(n)) - d(n), where d(n) is the number of divisors function.

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A060903 a(n) = floor(6nsqrt(n)/Pi^2).