A027601 -id:A027601 - cp's OEIS frontend

A001156 Number of partitions of n into squares.

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13, 14, 14, 16, 19, 20, 21, 23, 26, 27, 28, 31, 34, 37, 38, 43, 46, 49, 50, 55, 60, 63, 66, 71, 78, 81, 84, 90, 98, 104, 107, 116, 124, 132, 135, 144, 154, 163, 169, 178, 192, 201, 209, 220, 235, 247, 256

Offset: 0

N. J. A. Sloane

Number of partitions of n such that number of parts equal to k is multiple of k for all k. - Vladeta Jovovic, Aug 01 2004
Of course p_{4*square}(n)>0. In fact p_{4*square}(32n+28)=3 times p_{4*square}(8n+7) and p_{4*square}(72n+69) is even. These seem to be the only arithmetic properties the function p_{4*square(n)} possesses. Similar results hold for partitions into positive squares, distinct squares and distinct positive squares. - Michael David Hirschhorn, May 05 2005
The Heinz numbers of these partitions are given by A324588. - Gus Wiseman, Mar 09 2019

			p_{4*square}(23)=1 because 23 = 3^2 + 3^2 + 2^2 + 1^2 and there is no other partition of 23 into squares.
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 +...
such that the g.f. A(x) satisfies the identity [_Paul D. Hanna_]:
A(x) = 1/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)*(1-x^25)*...)
A(x) = 1 + x/(1-x) + x^4/((1-x)*(1-x^4)) + x^9/((1-x)*(1-x^4)*(1-x^9)) + x^16/((1-x)*(1-x^4)*(1-x^9)*(1-x^16)) + ...
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(14) = 6 integer partitions into squares are:
  (941)
  (911111)
  (44411)
  (44111111)
  (41111111111)
  (11111111111111)
while the a(14) = 6 integer partitions in which the multiplicity of k is a multiple of k for all k are:
  (333221)
  (33311111)
  (22222211)
  (2222111111)
  (221111111111)
  (11111111111111)
(End)

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).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301. (Annotated scanned copy)
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
M. D. Hirschhorn and J. A. Sellers, On a problem of Lehmer on partitions into squares, The Ramanujan Journal 8 (2004), 279-287.
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems, arXiv:math/0604019 [math.GM], 2006.
Eric Weisstein's World of Mathematics, Partition
Eric Weisstein's World of Mathematics, Smarandache Sequences
Eric Weisstein's World of Mathematics, Square Number

Cf. A000041, A000161 (partitions into 2 squares), A000290, A033461, A131799, A218494, A285218, A304046.
Cf. A078134 (first differences).
Cf. A003108, A037444, A046042, A259792, A259793, A294529.
Row sums of A243148.
Euler trans. of A010052 (see also A308297).
Cf. A001462, A003114, A006141, A011757, A039900, A047993, A052335, A062457, A064174, A078135, A109298, A117144.
Cf. A324524, A324572, A324587, A324588.

a001156 = p (tail a000290_list) where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Oct 31 2012, Aug 14 2011

m:=70; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^(k^2)): k in [1..(m+2)]]) )); // G. C. Greubel, Nov 11 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..120);  # Alois P. Heinz, May 30 2014

CoefficientList[ Series[Product[1/(1 - x^(m^2)), {m, 70}], {x, 0, 68}], x] (* Or *)
Join[{1}, Table[Length@PowersRepresentations[n, n, 2], {n, 68}]] (* Robert G. Wilson v, Apr 12 2005, revised Sep 27 2011 *)
f[n_] := Length@ IntegerPartitions[n, All, Range@ Sqrt@ n^2]; Array[f, 67] (* Robert G. Wilson v, Apr 14 2013 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2>n, 0, b[n-i^2, i]]]]; a[n_] := b[n, Sqrt[n]//Floor]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)

{a(n)=polcoeff(1/prod(k=1, sqrtint(n+1), 1-x^(k^2)+x*O(x^n)), n)} \\ Paul D. Hanna, Mar 09 2012

{a(n)=polcoeff(1+sum(m=1, sqrtint(n+1), x^(m^2)/prod(k=1, m, 1-x^(k^2)+x*O(x^n))), n)} \\ Paul D. Hanna, Mar 09 2012

G.f.: Product_{m>=1} 1/(1-x^(m^2)).
G.f.: Sum_{n>=0} x^(n^2) / Product_{k=1..n} (1 - x^(k^2)). - Paul D. Hanna, Mar 09 2012
a(n) = (1/n)*Sum_{k=1..n} A035316(k)*a(n-k). - Vladeta Jovovic, Nov 20 2002
a(n) = f(n,1,3) with f(x,y,z) = if xReinhard Zumkeller, Nov 08 2009
Conjecture (Jan Bohman, Carl-Erik Fröberg, Hans Riesel, 1979): a(n) ~ c * n^(-alfa) * exp(beta*n^(1/3)), where c = 1/18.79656, beta = 3.30716, alfa = 1.16022. - Vaclav Kotesovec, Aug 19 2015
From Vaclav Kotesovec, Dec 29 2016: (Start)
Correct values of these constants are:
1/c = sqrt(3) * (4*Pi)^(7/6) / Zeta(3/2)^(2/3) = 17.49638865935104978665...
alfa = 7/6 = 1.16666666666666666...
beta = 3/2 * (Pi/2)^(1/3) * Zeta(3/2)^(2/3) = 3.307411783596651987...
a(n) ~ 3^(-1/2) * (4*Pi*n)^(-7/6) * Zeta(3/2)^(2/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)). [Hardy & Ramanujan, 1917]
(End)

More terms from Eric W. Weisstein

More terms from Gh. Niculescu (ghniculescu(AT)yahoo.com), Oct 08 2006

A000048 Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 9, 16, 28, 51, 93, 170, 315, 585, 1091, 2048, 3855, 7280, 13797, 26214, 49929, 95325, 182361, 349520, 671088, 1290555, 2485504, 4793490, 9256395, 17895679, 34636833, 67108864, 130150493, 252645135, 490853403, 954437120, 1857283155

Offset: 0

N. J. A. Sloane

Also, for any m which is a multiple of n, the number of 2m-bead balanced binary necklaces of fundamental period 2n that are equivalent to their complements. [Clarified by Aaron Meyerowitz, Jun 01 2024]
Also binary Lyndon words of length n with an odd number of 1's (for n>=1).
Also number of binary irreducible polynomials of degree n having trace 1.
Also number of binary irreducible polynomials of degree n having linear coefficient 1 (this is the same as the trace-1 condition, as the reciprocal of an irreducible polynomial is again irreducible).
Also number of binary irreducible self-reciprocal polynomials of degree 2*n; there is no such polynomial for odd degree except for x+1.
Also number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 1 (mod n+1) = size of Varshamov-Tenengolts code VT_1(n).
Also the number of dynamical cycles of period 2n of a threshold Boolean automata network which is a quasi-minimal negative circuit of size nq where q is odd and which is updated in parallel. - Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009
Also the number of 3-elements orbits of the symmetric group S3 action on irreducible polynomials of degree 2n, n>1, over GF(2). - Jean Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Oct 04 2009
Conjecture: Also the number of caliber-n cycles of Zagier-reduced indefinite binary quadratic forms with sum invariant equal to s, where (s-1)/n is an odd integer. - Barry R. Smith, Dec 14 2014
The Metropolis, Stein, Stein (1973) reference on page 31 Table II lists a(k) for k = 2 to 15 and is actually for sequence A056303 since there a(k) = 0 for k<2. - Michael Somos, Dec 20 2014

			a(5) = 3 corresponding to the necklaces 00001, 00111, 01011.
a(6) = 5 from 000001, 000011, 000101, 000111, 001011.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
H. Kawakami, Table of rotation sequences of x_{n+1} = x_n^2 - lambda, pp. 73-92 of G. Ikegami, Editor, Dynamical Systems and Nonlinear Oscillations, Vol. 1, World Scientific, 1986.
Robert M. May, "Simple mathematical models with very complicated dynamics." Nature, Vol. 261, June 10, 1976, pp. 459-467; reprinted in The Theory of Chaotic Attractors, pp. 85-93. Springer, New York, NY, 2004. The sequences listed in Table 2 are A000079, A027375, A000031, A001037, A000048, A051841. - N. J. A. Sloane, Mar 17 2019
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).

G. C. Greubel, Table of n, a(n) for n = 0..3320 (terms 0..200 from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), p.408 and p.848.
L. Carlitz, A theorem of Dickson on irreducible polynomials, Proc. Amer. Math. Soc. 3, (1952). 693-700.
CombOS - Combinatorial Object Server, Generate necklaces, Lyndon words, chord diagrams, and relatives
J. Demongeot, M. Noual and S. Sene, On the number of attractors of positive and negative threshold Boolean automata circuits, hal-00647877 preprint (2009). [From Mathilde Noual (mathilde.noual(AT)ens-lyon.fr), Mar 03 2009]
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.
J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 2.
J. E. Iglesias, Enumeration of polytypes MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 178-194, Table 1.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
A. A. Kulkarni, N. Kiyavash and R. Sreenivas, On the Varshamov-Tenengolts Construction on Binary Strings, 2013.
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (Jun 10, 1976), 459-467.
N. Metropolis, M. L. Stein and P. R. Stein, On finite limit sets for transformations on the unit interval, J. Combin. Theory, A 15 (1973), 25-44; reprinted in P. Cvitanovic, ed., Universality in Chaos, Hilger, Bristol, 1986, pp. 187-206.
H. Meyn and W. Götz, Self-reciprocal polynomials over finite fields, Séminaire Lotharingien de Combinatoire, B21d (1989), 8 pp.
Simon Michalowsky, Bahman Gharesifard, Christian Ebenbauer, A Lie bracket approximation approach to distributed optimization over directed graphs, arXiv:1711.05486 [math.OC], 2017.
Tilman Piesk, Lists of the three sets of necklaces for n=1..12
R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
Frank Ruskey, Number of q-ary Lyndon words with given trace mod q
Frank Ruskey, Number of Lyndon words over GF(q) with given trace
N. J. A. Sloane, On single-deletion-correcting codes, arXiv:math/0207197 [math.CO], 2002; in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane, On single-deletion-correcting codes
B. R. Smith, Reducing quadratic forms by kneading sequences, Journal of Integer Sequences, 17 (2014), article 14.11.8.
P. R. Stein, Letter to N. J. A. Sloane, Jun 02 1971
J.-Y. Thibon, The cycle enumerator of unimodal permutations, arXiv:math/0102051 [math.CO], 2001.
Index entries for "core" sequences
Index entries for sequences related to Lyndon words
Index entries for sequences related to subset sums modulo m

Like A000013, but primitive necklaces. Half of A064355.
Equals A042981 + A042982.
Cf. A002823, A000016, A053633, A051841, A001037, A002075, A002076, A008683.
Cf. also A001037, A056303.
Very close to A006788 [Fisher, 1989].
bisection (odd terms) is A131203

with(numtheory); A000048 := proc(n) local d,t1; if n = 0 then RETURN(1) else t1 := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t1 := t1+mobius(d)*2^(n/d)/(2*n); fi; od; RETURN(t1); fi; end;

a[n_] := Total[ MoebiusMu[#]*2^(n/#)& /@ Select[ Divisors[n], OddQ]]/(2n); a[0] = 1; Table[a[n], {n,0,35}] (* Jean-François Alcover, Jul 21 2011 *)
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, MoebiusMu[#] 2^(n/#) &, OddQ] / (2 n)]; (* Michael Somos, Dec 20 2014 *)

A000048(n) = sumdiv(n,d,(d%2)*(moebius(d)*2^(n/d)))/(2*n) \\ Michael B. Porter, Nov 09 2009

L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%2==1, L(n, k), 0 ) ) / n;
vector(55,n,a(n)) \\ Joerg Arndt, Jun 28 2012

from sympy import divisors, mobius
def a(n): return 1 if n<1 else sum(mobius(d)*2**(n//d) for d in divisors(n) if d%2)//(2*n) # Indranil Ghosh, Apr 28 2017

a(n) = (1/(2*n)) * Sum_{odd d divides n} mu(d)*2^(n/d), where mu is the Mobius function A008683.
a(n) = A056303(n) for all integer n>=2. - Michael Somos, Dec 20 2014
Sum_{k dividing m for which m/k is odd} k*a(k) = 2^(m-1). (This explains the observation that the sequence is very close to A006788. Unless m has some nontrivial odd divisors that are small relative to m, the term m*a(m) will dominate the sum. Thus, we see for instance that a(n) = A006788(n) when n has one of the forms 2^m or 2^m*p where p is an odd prime with a(2^m) < p.) - Barry R. Smith, Oct 24 2015
A000013(n) = Sum_{d|n} a(d). - Robert A. Russell, Jun 09 2019
G.f.: 1 + Sum_{k>=1} mu(2*k)*log(1 - 2*x^k)/(2*k). - Ilya Gutkovskiy, Nov 11 2019

Additional comments from Frank Ruskey, Dec 13 1999

A037444 Number of partitions of n^2 into squares.

Original entry on oeis.org

1, 1, 2, 4, 8, 19, 43, 98, 220, 504, 1116, 2468, 5368, 11592, 24694, 52170, 108963, 225644, 462865, 941528, 1899244, 3801227, 7550473, 14889455, 29159061, 56722410, 109637563, 210605770, 402165159, 763549779, 1441686280, 2707535748, 5058654069, 9404116777

Offset: 0

Wouter Meeussen

Is lim_{n->inf} a(n)^(1/n) > 1? - Paul D. Hanna, Aug 20 2002

The limit above is equal to 1 (see formula by Hardy & Ramanujan for A001156). - Vaclav Kotesovec, Dec 29 2016

T. D. Noe, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..945 (terms n = 0..100 from T. D. Noe, terms n = 101..500 from Alois P. Heinz)
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

Entries with square index in A001156.
Cf. A072964, A030273, A000041, A000290, A229239, A229468.
Cf. A003108, A046042.
Cf. A259792, A259793.
A row or column of the array in A259799.

a037444 n = p (map (^ 2) [1..]) (n^2) where
   p _      0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 14 2011

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 15 2013

max=33; se = Series[ Product[1/(1-x^(k^2)), {k, 1, max}], {x, 0, max^2}]; a[n_] := Coefficient[se, x^(n^2)]; a[0] = 1; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Oct 18 2011 *)

a(n) = A001156(n^2) = coefficient of x^(n^2) in the series expansion of Prod_{k>=1} 1/(1 - x^(k^2)).

a(n) ~ 3^(-1/2) * (4*Pi)^(-7/6) * Zeta(3/2)^(2/3) * n^(-7/3) * exp(2^(-4/3) * 3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3)) [Hardy & Ramanujan, 1917, modified from A001156]. - Vaclav Kotesovec, Dec 29 2016

A259792 Number of partitions of n^3 into cubes.

Original entry on oeis.org

1, 1, 2, 5, 17, 62, 258, 1050, 4365, 18012, 73945, 301073, 1214876, 4852899, 19187598, 75070201, 290659230, 1113785613, 4224773811, 15866483556, 59011553910, 217410395916, 793635925091, 2871246090593, 10297627606547, 36620869115355, 129166280330900

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..173 (terms 0..120 from Alois P. Heinz)
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

A row of the array in A259799.
Cf. A279329.
Cf. A001156, A003108, A046042.
Cf. A037444, A259793.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^3>n, 0, b(n-i^3, i)))
    end:
a:= n-> b(n^3, n):
seq(a(n), n=0..26);  # Alois P. Heinz, Jul 10 2015

$RecursionLimit = 1000; b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[ i^3>n, 0, b[n-i^3, i]]]; a[n_] := b[n^3, n]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

a(n) = [x^(n^3)] Product_{j>=1} 1/(1-x^(j^3)). - Alois P. Heinz, Jul 10 2015
a(n) = A003108(n^3). - Vaclav Kotesovec, Aug 19 2015
a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * n^(3/4) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(15/4)) [after Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

More term from Alois P. Heinz, Jul 10 2015

A259793 Number of partitions of n^4 into fourth powers.

Original entry on oeis.org

1, 1, 2, 7, 36, 253, 1886, 14800, 118238, 955639, 7750456, 62777522, 506272363, 4056634991, 32252971687, 254209569990, 1985108901344, 15352968310930, 117579612410477, 891596419221856, 6694250497509934, 49768995849050468, 366423320400440927, 2671969175372760210

Offset: 0

N. J. A. Sloane, Jul 06 2015

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..63 (terms 0..45 from Alois P. Heinz)
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.

A row of the array in A259799.
Cf. A001156, A003108, A046042.
Cf. A037444, A259792.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1) +`if`(i^4>n, 0, b(n-i^4, i)))
    end:
a:= n-> b(n^4, n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 10 2015

$RecursionLimit = 10^4; b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i^4>n, 0, b[n-i^4, i]]]; a[n_] := b[n^4, n];  Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)

a(n) = [x^(n^4)] Product_{j>=1} 1/(1-x^(j^4)). - Alois P. Heinz, Jul 10 2015
a(n) = A046042(n^4). - Vaclav Kotesovec, Aug 19 2015
a(n) ~ exp(5 * (Gamma(1/4)*Zeta(5/4))^(4/5) * n^(4/5) / 2^(16/5)) * (Gamma(1/4)*Zeta(5/4))^(4/5) / (2^(47/10) * sqrt(5) * Pi^(5/2) * n^(26/5)) [after Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

More terms from Alois P. Heinz, Jul 10 2015

A259799 Array read by antidiagonals upwards: T(n,k) = number of partitions of k^n into n-th powers (n>=1, k>=0).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 5, 8, 7, 1, 1, 2, 7, 17, 19, 11, 1, 1, 2, 9, 36, 62, 43, 15, 1, 1, 2, 13, 88, 253, 258, 98, 22, 1, 1, 2, 19, 218, 1104, 1886, 1050, 220, 30, 1, 1, 2, 27, 550, 5082, 15772, 14800, 4365, 504, 42, 1, 1, 2, 40, 1413, 24119, 140549, 241582, 118238, 18012, 1116, 56

Offset: 1

N. J. A. Sloane, Jul 06 2015

			The array begins:
  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, ...
  1, 1, 2, 4, 8, 19, 43, 98, 220, 504, ...
  1, 1, 2, 5, 17, 62, 258, 1050, 4365, 18012, ...
  1, 1, 2, 7, 36, 253, 1886, 14800, 118238, ...
  1, 1, 2, 9, 88, 1104, 15772, 241582, ...
  ...

Alois P. Heinz, Antidiagonals n = 1..16, flattened
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989

Rows: A000041, A037444, A259792-A259795.
Columns: A259796, A027601, A259797, A259798.
T(n,n) gives A331402.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i=2, 1+iquo(n, i^k), b(n, i-1, k)+
      `if`(i^k>n, 0, b(n-i^k, i, k))))
    end:
T:= (n, k)-> b(k^n, k, n):
seq(seq(T(d-k, k), k=0..d-1), d=1..12);  # Alois P. Heinz, Jul 10 2015

b[n_, i_, k_] := b[n, i, k] = If[n==0 || i==1, 1, If[i==2, 1+Quotient[n, i^k], b[n, i-1, k] + If[i^k>n, 0, b[n-i^k, i, k]]]]; T[n_, k_] := b[k^n, k, n]; Table[ Table[ T[d-k, k], {k, 0, d-1}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jul 10 2015

A006788 a(n) = floor(2^(n-1)/n).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 28, 51, 93, 170, 315, 585, 1092, 2048, 3855, 7281, 13797, 26214, 49932, 95325, 182361, 349525, 671088, 1290555, 2485513, 4793490, 9256395, 17895697, 34636833, 67108864, 130150524, 252645135, 490853405, 954437176, 1857283155, 3616814565

Offset: 1

N. J. A. Sloane

nonn

Very close to A000048. [Fisher, 1989]

This is the number of nested polygons needed to produce a graph that is always concave, see the MathWorld article. - Jon Perry, Sep 15 2002

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
H. L. Fisher, Letter to N. J. A. Sloane, Mar 16 1989
Simon Michalowsky, Bahman Gharesifard and Christian Ebenbauer, A Lie bracket approximation approach to distributed optimization over directed graphs, arXiv:1711.05486 [math.OC], 2017.
Eric Weisstein's World of Mathematics, Happy End Problem

Cf. A054650, A000048.

[Floor(2^(n-1)/n) : n in [1..40]]; // Vincenzo Librandi, Sep 24 2011

Table[Quotient[2^n, 2*n], {n, 1, 60}] (* Vladimir Joseph Stephan Orlovsky, May 07 2011 *)

print([2**(n-1)//n for n in range(1, 40)]) # Gennady Eremin, Feb 04 2022

A006788 = lambda n: (1<A006788(n) for n in (1..38)] # Peter Luschny, Sep 18 2014

cp's OEIS Frontend

A001156 Number of partitions of n into squares.

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A000048 Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged.

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A037444 Number of partitions of n^2 into squares.

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A259792 Number of partitions of n^3 into cubes.

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A259793 Number of partitions of n^4 into fourth powers.

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A259799 Array read by antidiagonals upwards: T(n,k) = number of partitions of k^n into n-th powers (n>=1, k>=0).

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