A063834
Twice partitioned numbers: the number of ways a number can be partitioned into not necessarily different parts and each part is again so partitioned.
Original entry on oeis.org
1, 1, 3, 6, 15, 28, 66, 122, 266, 503, 1027, 1913, 3874, 7099, 13799, 25501, 48508, 88295, 165942, 299649, 554545, 997281, 1817984, 3245430, 5875438, 10410768, 18635587, 32885735, 58399350, 102381103, 180634057, 314957425, 551857780, 958031826, 1667918758
Offset: 0
G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + 266*x^8 + ...
If n=6, a possible first partitioning is (3+3), resulting in the following second partitionings: ((3),(3)), ((3),(2+1)), ((3),(1+1+1)), ((2+1),(3)), ((2+1),(2+1)), ((2+1),(1+1+1)), ((1+1+1),(3)), ((1+1+1),(2+1)), ((1+1+1),(1+1+1)).
A001970 counts multiset partitions of integer partitions.
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1)+`if`(i>n, 0, numbpart(i)*b(n-i, i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Nov 26 2015
-
Table[Plus @@ Apply[Times, IntegerPartitions[i] /. i_Integer :> PartitionsP[i], 2], {i, 36}]
(* second program: *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + If[i > n, 0, PartitionsP[i]*b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
-
{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 19 2016 */
A001970
Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.
Original entry on oeis.org
1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876, 203286806, 340435588, 568496753, 946695386
Offset: 0
G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + ...
a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).
The a(4)=14 multiset partitions whose total sum of parts is 4 are:
((4)),
((13)), ((1)(3)),
((22)), ((2)(2)),
((112)), ((1)(12)), ((2)(11)), ((1)(1)(2)),
((1111)), ((1)(111)), ((11)(11)), ((1)(1)(11)), ((1)(1)(1)(1)). - _Gus Wiseman_, Dec 19 2016
- A. Cayley, Recherches sur les matrices dont les termes sont des fonctions linéaires d'une seule indéterminée, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
- V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
- 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).
- J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
- J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order', Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.
- Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 (first 500 terms from T. D. Noe)
- Pieter Belmans, Segre symbols, 2016.
- Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See pp. 10, 16.
- P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
- P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148
- R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).
- L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
- M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, 2014.
- M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
- XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, Science China Physics, Mechanics & Astronomy, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9.
- Jessie Pitsillides, Segre Characteristic Equivalence, arXiv:2506.12065 [math.GM], 2025.
- Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
- N. J. A. Sloane, Transforms.
- N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
- J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
- Index entries for sequences related to rooted trees
Related to
A001383 via generating function.
The multiplicative version (factorizations) is
A050336.
The ordered version (sequences of partitions) is
A055887.
-
Following Vladeta Jovovic:
a001970 n = a001970_list !! (n-1)
a001970_list = 1 : f 1 [1] where
f x ys = y : f (x + 1) (y : ys) where
y = sum (zipWith (*) ys a061259_list) `div` x
-- Reinhard Zumkeller, Oct 31 2015
-
with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
# second Maple program:
with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
numbpart(d), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..35); # Alois P. Heinz, Dec 19 2016
-
m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* Jean-François Alcover, Jul 19 2011, after g.f. *)
-
{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k + x * O(x^n)), n))}; /* Michael Somos, Dec 20 2016 */
-
from sympy.core.cache import cacheit
from sympy import npartitions, divisors
@cacheit
def a(n): return 1 if n == 0 else sum([sum([d*npartitions(d) for d in divisors(j)])*a(n - j) for j in range(1, n + 1)]) / n
[a(n) for n in range(51)] # Indranil Ghosh, Aug 19 2017, after Maple code
# (Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 1)
a = EulerTransform(EulerTransform(b))
print([a(n) for n in range(36)]) # Peter Luschny, Nov 17 2022
A270995
Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).
Original entry on oeis.org
1, 1, 2, 4, 7, 12, 23, 37, 64, 108, 180, 290, 488, 772, 1251, 2001, 3180, 4982, 7913, 12261, 19162, 29669, 45804, 70187, 108029, 164276, 250267, 379439, 574067, 864044, 1302169, 1949050, 2917900, 4352796, 6481627, 9620256, 14274080, 21090608, 31142909
Offset: 0
a(6)=23: {(6), (5)(1), (51), (4)(2), (42), (4)(1)(1), (41)(1), (3)(3), (3)(2)(1), (3)(21), (32)(1), (31)(2), (21)(3), (321), (3)(1)(1)(1), (31)(1)(1), (2)(2)(2), (2)(2)(1)(1), (21)(2)(1), (21)(21), (2)(1)(1)(1)(1), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1)}.
For compositions instead of partitions we have
A304969, non-strict
A055887.
A072233 counts partitions by sum and length.
-
nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
A055887
Number of ordered partitions of partitions.
Original entry on oeis.org
1, 1, 3, 8, 22, 59, 160, 431, 1164, 3140, 8474, 22864, 61697, 166476, 449210, 1212113, 3270684, 8825376, 23813776, 64257396, 173387612, 467856828, 1262431711, 3406456212, 9191739970, 24802339472, 66924874539, 180585336876, 487278670744, 1314838220172
Offset: 0
The a(4) = 22 chains of multisets, where notation x-y means "y is a submultiset of x", are: (o-o-o-o) (oo-o-o) (oo-oo) (ooo-o) (oooo) (oe-o-o) (ooe-o) (oooe) (oe-oe) (ooe-e) (oee-o) (ooee) (oei-o) (ooei) (oe-e-e) (oee-e) (oeee) (oei-e) (oeei) (oei-i) (oeii) (oeis).
From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose an integer partition of each part of an integer composition of n. The a(0) = 1 through a(3) = 8 choices are:
() ((1)) ((2)) ((3))
((11)) ((21))
((1)(1)) ((111))
((1)(2))
((2)(1))
((1)(11))
((11)(1))
((1)(1)(1))
(End)
A011782 counts integer compositions.
A072233 counts partitions by sum and length.
-
with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
P := (x) -> product( 1/(1-x^k), k=1..20 ) - 1; F := (x) -> series( 1/(1-P(x)) - 1, x, 21 ); # F(x) is g.f. for this sequence # Warren D. Smith, Jan 28 2002
A055887rec:= proc(n::integer) local k; option remember; with(combinat): if n = 0 then 1 else add(numbpart(k) *procname(n - k), k=1..n); end if; end proc: seq (A055887rec(n), n=0..10); # Thomas Wieder, Nov 26 2007
-
a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* Geoffrey Critzer, Dec 23 2010 *)
(1/(2 - 1/QPochhammer[x]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
Table[Sum[Times@@PartitionsP/@c,{c,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}] (* Gus Wiseman, Jul 31 2022 *)
-
Vec(1/(2-1/eta(x+O(x^66)))) \\ Joerg Arndt, Sep 30 2012
A023893
Number of partitions of n into prime power parts (1 included); number of nonisomorphic Abelian subgroups of symmetric group S_n.
Original entry on oeis.org
1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 105, 134, 171, 215, 269, 335, 415, 511, 626, 764, 929, 1125, 1356, 1631, 1953, 2333, 2776, 3296, 3903, 4608, 5427, 6377, 7476, 8744, 10205, 11886, 13818, 16032, 18565, 21463, 24768, 28536
Offset: 0
From _Gus Wiseman_, Jul 28 2022: (Start)
The a(0) = 1 through a(6) = 10 partitions:
() (1) (2) (3) (4) (5) (33)
(11) (21) (22) (32) (42)
(111) (31) (41) (51)
(211) (221) (222)
(1111) (311) (321)
(2111) (411)
(11111) (2211)
(3111)
(21111)
(111111)
(End)
The multiplicative version (factorizations) is
A000688.
The version for just primes (not prime-powers) is
A034891, strict
A036497.
These partitions are ranked by
A302492.
A001222 counts prime-power divisors.
A072233 counts partitions by sum and length.
-
Table[Length[Select[IntegerPartitions[n],Count[Map[Length,FactorInteger[#]], 1] == Length[#] &]], {n, 0, 35}] (* Geoffrey Critzer, Oct 25 2015 *)
nmax = 50; Clear[P]; P[m_] := P[m] = Product[Product[1/(1-x^(p^k)), {k, 1, m}], {p, Prime[Range[PrimePi[nmax]]]}]/(1-x)+O[x]^nmax // CoefficientList[ #, x]&; P[1]; P[m=2]; While[P[m] != P[m-1], m++]; P[m] (* Jean-François Alcover, Aug 31 2016 *)
-
lista(m) = {x = t + t*O(t^m); gf = prod(k=1, m, if (isprimepower(k), 1/(1-x^k), 1))/(1-x); for (n=0, m, print1(polcoeff(gf, n, t), ", "));} \\ Michel Marcus, Mar 09 2013
-
from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A023893(n):
@lru_cache(maxsize=None)
def c(n): return sum((p**(e+1)-p)//(p-1) for p,e in factorint(n).items())+1
return (c(n)+sum(c(k)*A023893(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024
A023894
Number of partitions of n into prime power parts (1 excluded).
Original entry on oeis.org
1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 23, 29, 37, 44, 54, 66, 80, 96, 115, 138, 165, 196, 231, 275, 322, 380, 443, 520, 607, 705, 819, 950, 1099, 1268, 1461, 1681, 1932, 2214, 2533, 2898, 3305, 3768, 4285, 4872, 5530, 6267, 7094, 8022, 9060
Offset: 0
From _Gus Wiseman_, Jul 28 2022: (Start)
The a(0) = 1 through a(9) = 7 partitions:
() . (2) (3) (4) (5) (33) (7) (8) (9)
(22) (32) (42) (43) (44) (54)
(222) (52) (53) (72)
(322) (332) (333)
(422) (432)
(2222) (522)
(3222)
(End)
The multiplicative version (factorizations) is
A000688, coprime
A354911.
Twice-partitions of this type are counted by
A279784, factorizations
A295935.
These partitions are ranked by
A355743.
A001222 counts prime-power divisors.
A072233 counts partitions by sum and length.
-
Table[Length[Select[IntegerPartitions[n],And@@PrimePowerQ/@#&]],{n,0,30}] (* Gus Wiseman, Jul 28 2022 *)
-
is_primepower(n)= {ispower(n, , &n); isprime(n)}
lista(m) = {x = t + t*O(t^m); gf = prod(k=1, m, if (is_primepower(k), 1/(1-x^k), 1)); for (n=0, m, print1(polcoeff(gf, n, t), ", "));}
\\ Michel Marcus, Mar 09 2013
-
from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A023894(n):
@lru_cache(maxsize=None)
def c(n): return sum((p**(e+1)-p)//(p-1) for p,e in factorint(n).items())
return (c(n)+sum(c(k)*A023894(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024
A107742
G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).
Original entry on oeis.org
1, 1, 2, 4, 6, 10, 17, 25, 38, 59, 86, 125, 184, 260, 369, 524, 726, 1005, 1391, 1894, 2576, 3493, 4687, 6272, 8373, 11090, 14647, 19294, 25265, 32991, 42974, 55705, 72025, 92895, 119349, 152965, 195592, 249280, 316991, 402215, 508932, 642598, 809739, 1017850, 1276959, 1599015, 1997943, 2491874, 3102477, 3855165, 4782408, 5922954
Offset: 0
A072233 counts partitions by sum and length.
-
nmax = 50; CoefficientList[Series[Product[(1+x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 28 2018 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@chQ/@#&]],{n,0,5}] (* Gus Wiseman, Sep 13 2022 *)
-
a(n)=polcoeff(prod(k=1,n,prod(j=1,n\k,1+x^(j*k)+x*O(x^n))),n) /* Paul D. Hanna */
-
N=66; x='x+O('x^N); gf=1/prod(j=0,N, eta(x^(2*j+1))); gf=prod(j=1,N,(1+x^j)^numdiv(j)); Vec(gf) /* Joerg Arndt, May 03 2008 */
-
{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^(2*m)+x*O(x^n))/m)),n))} /* Paul D. Hanna, Mar 28 2009 */
A006951
Number of conjugacy classes in GL(n,2).
Original entry on oeis.org
1, 1, 3, 6, 14, 27, 60, 117, 246, 490, 1002, 1998, 4053, 8088, 16284, 32559, 65330, 130626, 261726, 523374, 1047690, 2095314, 4192479, 8384808, 16773552, 33546736, 67101273, 134202258, 268420086, 536839446, 1073710914, 2147420250, 4294904430, 8589807438
Offset: 0
For the 5 partitions of 4 (namely [1^4]; [2,1^2]; [2^2]; [3,1]; [4]) we have
(f(m) = 2^(m-1)*(2-1) = 2^(m-1) and)
f([1^4]) = 2^3 = 8,
f([2,1^2]) = 1*2^1 = 2,
f([2^2]) = 2^1 = 2,
f([3,1]) = 1*1 = 1,
f([4]) = 1,
the sum is 8+2+2+1+1 = 14 = a(4).
- _Joerg Arndt_, Jan 02 2013
- W. D. Smith, personal communication.
- 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..1000
- W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 161
- I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bulletin of the Australian Mathematical Society, vol.23, no.01, pp.23-48, (February-1981).
- N. J. A. Sloane, Transforms
-
/* The program does not work for n>19: */
[1] cat [NumberOfClasses(GL(n,2)): n in [1..19]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi Jan 24 2013
-
with(numtheory):
b:= n-> add(phi(d)*2^(n/d), d=divisors(n))/n-1:
a:= proc(n) option remember; `if`(n=0, 1,
add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Oct 20 2012
-
b[n_] := Sum[EulerPhi[d]*2^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
Table[Sum[2^(Length[ptn]-Length[Split[ptn]]),{ptn,IntegerPartitions[n]}],{n,30}] (* Gus Wiseman, Jan 21 2019 *)
-
N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-2*x^n) );
v=Vec(gf)
/* Joerg Arndt, Jan 02 2013 */
A304969
Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).
Original entry on oeis.org
1, 1, 2, 5, 11, 25, 57, 129, 292, 662, 1500, 3398, 7699, 17443, 39519, 89536, 202855, 459593, 1041267, 2359122, 5344889, 12109524, 27435660, 62158961, 140828999, 319065932, 722884274, 1637785870, 3710611298, 8406859805, 19046805534, 43152950024, 97768473163
Offset: 0
From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose a strict integer partition of each part of an integer composition of n. The a(1) = 1 through a(4) = 11 choices are:
((1)) ((2)) ((3)) ((4))
((1)(1)) ((21)) ((31))
((1)(2)) ((1)(3))
((2)(1)) ((2)(2))
((1)(1)(1)) ((3)(1))
((1)(21))
((21)(1))
((1)(1)(2))
((1)(2)(1))
((2)(1)(1))
((1)(1)(1)(1))
(End)
For partitions instead of compositions we have
A270995, non-strict
A063834.
A072233 counts partitions by sum and length.
-
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
a:= proc(n) option remember; `if`(n=0, 1,
add(b(j)*a(n-j), j=1..n))
end:
seq(a(n), n=0..40); # Alois P. Heinz, May 22 2018
-
nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]
A381636
Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.
Original entry on oeis.org
12, 60, 63, 84, 120, 126, 132, 156, 204, 228, 252, 276, 300, 315, 325, 348, 372, 420, 444, 492, 504, 516, 560, 564, 588, 630, 636, 650, 660, 693, 708, 720, 732, 780, 804, 819, 840, 852, 876, 924, 931, 948, 975, 996, 1008, 1020, 1068, 1071, 1092, 1140, 1164
Offset: 1
The prime indices of 300 are {1,1,2,3,3}, with partitions into constant blocks:
{{2},{1,1},{3,3}}
{{1},{1},{2},{3,3}}
{{2},{3},{3},{1,1}}
{{1},{1},{2},{3},{3}}
but none of these has distinct block-sums, so 300 is in the sequence.
The terms together with their prime indices begin:
12: {1,1,2}
60: {1,1,2,3}
63: {2,2,4}
84: {1,1,2,4}
120: {1,1,1,2,3}
126: {1,2,2,4}
132: {1,1,2,5}
156: {1,1,2,6}
204: {1,1,2,7}
228: {1,1,2,8}
252: {1,1,2,2,4}
276: {1,1,2,9}
300: {1,1,2,3,3}
These are the positions of 0 in
A381635, after taking block-sums
A381716.
Partitions of this type are counted by
A381717.
For strict instead of constant blocks we have
A381806, zeros of
A381633.
For equal instead of distinct block-sums we have
A381871.
A050361 counts multiset partitions into distinct constant blocks, after sums
A381715.
Cf.
A000720,
A001222,
A005117,
A050320,
A059404,
A213242,
A293243,
A299202,
A300385,
A381078,
A381454,
A381634.
-
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Select[pfacs[#],UnsameQ@@hwt/@#&]=={}&]
Showing 1-10 of 41 results.
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