A226748 -id:A226748 - cp's OEIS frontend

A003108 Number of partitions of n into cubes.

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 21, 21, 21, 22, 22, 22, 23, 23, 25, 26, 26, 27, 27, 27, 28

Offset: 0

N. J. A. Sloane, Herman P. Robinson

nonn

The g.f. 1/(z+1)/(z**2+1)/(z**4+1)/(z-1)**2 conjectured by Simon Plouffe in his 1992 dissertation is wrong.

			a(16) = 3 because we have [8,8], [8,1,1,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 +...
such that the g.f. A(x) satisfies the identity [Paul D. Hanna]:
A(x) = 1/((1-x)*(1-x^8)*(1-x^27)*(1-x^64)*(1-x^125)*...)
A(x) = 1 + x/(1-x) + x^8/((1-x)*(1-x^8)) + x^27/((1-x)*(1-x^8)*(1-x^27)) + x^64/((1-x)*(1-x^8)*(1-x^27)*(1-x^64)) +...

H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..100000 (terms 0..1000 from T. D. Noe)
G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proceedings of the London Mathematical Society, 2, XVI, 1917, p. 373.
F. Iacobescu, Smarandache Partition Type and Other Sequences, Bull. Pure Appl. Sci. 16E, 237-240, 1997.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Cubic Number
Eric Weisstein's World of Mathematics, Partition
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A000578, A068980, A131799, A218495, A226748, A279329, A280263.
Cf. A001156, A046042.
Cf. A037444, A259792, A259793.

a003108 = p $ tail a000578_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

[#RestrictedPartitions(n,{d^3:d in [1..n]}): n in [0..150]]; // Marius A. Burtea, Jan 02 2019

g:=1/product(1-x^(j^3),j=1..30): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..65); # Emeric Deutsch, Mar 30 2006

nmax = 100; CoefficientList[Series[Product[1/(1 - x^(k^3)), {k, 1, nmax^(1/3)}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 60; cmax = nmax^(1/3);
s = Table[n^3, {n, cmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)

{a(n)=polcoeff(1/prod(k=1, ceil(n^(1/3)), 1-x^(k^3)+x*O(x^n)), n)} /* Paul D. Hanna, Mar 09 2012 */

{a(n)=polcoeff(1+sum(m=1, ceil(n^(1/3)), x^(m^3)/prod(k=1, m, 1-x^(k^3)+x*O(x^n))), n)} /* Paul D. Hanna, Mar 09 2012 */

from functools import lru_cache
from sympy import integer_nthroot, divisors
@lru_cache(maxsize=None)
def A003108(n):
    @lru_cache(maxsize=None)
    def a(n): return integer_nthroot(n,3)[1]
    @lru_cache(maxsize=None)
    def c(n): return sum(d for d in divisors(n,generator=True) if a(d))
    return (c(n)+sum(c(k)*A003108(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

G.f.: 1/Product_{j>=1} (1-x^(j^3)). - Emeric Deutsch, Mar 30 2006
G.f.: Sum_{n>=0} x^(n^3) / Product_{k=1..n} (1 - x^(k^3)). - Paul D. Hanna, Mar 09 2012
a(n) ~ exp(4 * (Gamma(1/3)*Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3)*Zeta(4/3))^(3/4) / (24*Pi^2*n^(5/4)) [Hardy & Ramanujan, 1917]. - Vaclav Kotesovec, Dec 29 2016

A068980 Number of partitions of n into nonzero tetrahedral numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 11, 11, 12, 12, 15, 15, 16, 16, 19, 19, 22, 22, 25, 25, 28, 29, 32, 32, 35, 36, 42, 42, 45, 46, 52, 53, 56, 57, 63, 64, 70, 71, 77, 78, 84, 87, 94, 95, 101, 104, 115, 116, 122, 125, 136, 139, 146, 149, 160, 163, 175

Offset: 0

Franklin T. Adams-Watters, Apr 01 2002

			a(10) = 4 because we can write 10 = 10 = 4 + 4 + 1 + 1 = 4 + 1 + 1 + 1 + 1 + 1 = 1 + ... + 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Zhicheng Gao, Andrew MacFie and Daniel Panario, Counting words by number of occurrences of some patterns, The Electronic Journal of Combinatorics, 18 (2011), #P143.

See also A007294 (partitions into triangular numbers), A000292 (tetrahedral numbers).

Cf. A003108, A226748.

nmax = 100; CoefficientList[Series[Product[1/(1-x^(k*(k+1)*(k+2)/6)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

G.f.: 1 / Product_{k>=3} (1 - z^binomial(k, 3)).

G.f.: Sum_{i>=0} x^(i*(i+1)*(i+2)/6) / Product_{j=1..i} (1 - x^(j*(j+1)*(j+2)/6)). - Ilya Gutkovskiy, Jun 08 2017

A053012 Platonic numbers: a(n) is a tetrahedral (A000292), cube (A000578), octahedral (A005900), dodecahedral (A006566) or icosahedral (A006564) number.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 19, 20, 27, 35, 44, 48, 56, 64, 84, 85, 120, 124, 125, 146, 165, 216, 220, 231, 255, 286, 343, 344, 364, 455, 456, 489, 512, 560, 670, 680, 729, 742, 816, 891, 969, 1000, 1128, 1140, 1156, 1330, 1331, 1469, 1540, 1629, 1728, 1771, 1834

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 22 2000

19, the 3rd octahedral number, is the only prime platonic number. - Jean-François Alcover, Oct 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Platonic numbers

Numbers of partitions into Platonic numbers: A226748, A226749.

a053012 n = a053012_list !! (n-1)
a053012_list = tail $ f
   [a000292_list, a000578_list, a005900_list, a006566_list, a006564_list]
   where f pss = m : f (map (dropWhile (<= m)) pss)
                 where m = minimum (map head pss)
-- Reinhard Zumkeller, Jun 17 2013

nn = 25; t1 = Table[n (n + 1) (n + 2)/6, {n, nn}]; t2 = Table[n^3, {n, nn}]; t3 = Table[(2*n^3 + n)/3, {n, nn}]; t4 = Table[n (3*n - 1) (3*n - 2)/2, {n, nn}]; t5 = Table[n (5*n^2 - 5*n + 2)/2, {n, nn}]; Select[Union[t1, t2, t3, t4, t5], # <= t1[[-1]] &] (* T. D. Noe, Oct 13 2012 *)

listpoly(lim, poly[..])=my(v=List()); for(i=1,#poly, my(P=poly[i], x=variable(P), f=k->subst(P,x,k),n,t); while((t=f(n++))<=lim, listput(v, t))); Set(v)
list(lim)=my(n='n); listpoly(lim, n*(n+1)*(n+2)/6, n^3, (2*n^3+n)/3, n*(3*n-1)*(3*n-2)/2, n*(5*n^2-5*n+2)/2) \\ Charles R Greathouse IV, Oct 11 2016

A226749 Number of partitions of n into distinct Platonic numbers, cf. A053012.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 7, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 11, 11, 11, 12, 13, 13, 12, 12, 13, 15, 15, 16, 17, 17, 16, 18, 18, 19, 19, 21, 21, 23, 24, 25, 24, 24, 24, 26, 26, 29, 32

Offset: 0

Reinhard Zumkeller, Jun 17 2013

nonn

			First Platonic numbers: 1, 4, 6, 8, 10, 12, 19, 20, 27, ...
a(10) = #{10, 6+4} = 2;
a(11) = #{10+1, 6+4+1} = 2;
a(12) = #{12, 8+4} = 2;
a(13) = #{12+1, 8+4+1} = 2;
a(14) = #{10+4, 8+6} = 2;
a(15) = #{10+4+1, 8+6+1} = 2;
a(16) = #{12+4, 10+6} = 2;
a(17) = #{12+4+1, 10+6+1} = 2;
a(18) = #{12+6, 10+8, 8+6+4} = 3;
a(19) = #{19, 12+6+1, 10+8+1, 8+6+4+1} = 4;
a(20) = #{20, 19+1, 12+8, 10+6+4} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A053012, A226748.

a226749 = p a053012_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

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A003108 Number of partitions of n into cubes.

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A068980 Number of partitions of n into nonzero tetrahedral numbers.

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A053012 Platonic numbers: a(n) is a tetrahedral (A000292), cube (A000578), octahedral (A005900), dodecahedral (A006566) or icosahedral (A006564) number.

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A226749 Number of partitions of n into distinct Platonic numbers, cf. A053012.

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