A296131 -id:A296131

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A107742

G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

+10
57

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

From Gus Wiseman, Sep 13 2022: (Start)

Also the number of multiset partitions of integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:

{{1},{1}} {{1,2}} {{1},{3}}

{{1},{2}} {{2},{2}}

{{1},{1},{1}} {{1},{1,2}}

{{1},{1},{2}}

{{1},{1},{1},{1}}

Intervals are counted by A001227, ranked by A073485.

The initial version is A007294.

The strict version is A327731.

The version for gapless multisets instead of intervals is A356941.

The case of strict partitions is A356957.

Also the number of multiset partitions of integer partitions of n into distinct constant blocks. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:

{{1},{2}} {{1},{3}}

{{1},{1,1}} {{1,1,1,1}}

{{2},{1,1}}

{{1},{1,1,1}}

Constant multisets are counted by A000005, ranked by A000961.

The non-strict version is A006171.

The unlabeled version is A089259.

The non-constant block version is A261049.

The version for twice-partitions is A279786, factorizations A296131.

Also the number of multiset partitions of integer partitions of n into constant blocks of odd length. For example, a(1) = 1 through a(4) = 6 multiset partitions are:

{{1},{1}} {{1,1,1}} {{1},{3}}

{{1},{2}} {{2},{2}}

{{1},{1},{1}} {{1},{1,1,1}}

{{1},{1},{2}}

{{1},{1},{1},{1}}

The strict version is A327731 (also).

(End)

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

N. J. A. Sloane, Transforms

FORMULA

Euler transform of A001227.

Weigh transform of A000005.

G.f. satisfies: log(A(x)) = Sum_{n>=1} A109386(n)/n*x^n, where A109386(n) = Sum_{d|n} d*Sum_{m|d} (m mod 2). - Paul D. Hanna, Jun 26 2005

G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^(2n)) /n ). - Paul D. Hanna, Mar 28 2009

G.f.: Product_{n>=1} Q(x^n) where Q(x) is the g.f. of A000009. - Joerg Arndt, Feb 27 2014

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 04 2017

Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/6). - Vaclav Kotesovec, Aug 29 2018

MATHEMATICA

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 *)

PROG

(PARI) a(n)=polcoeff(prod(k=1, n, prod(j=1, n\k, 1+x^(j*k)+x*O(x^n))), n) /* Paul D. Hanna */

(PARI) 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 */

(PARI) {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 */

CROSSREFS

Cf. A006171, A109386, A219554, A280473, A280486, A288007.

Product_{k>=1} (1 + x^k)^sigma_m(k): this sequence (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

A000041 counts integer partitions, strict A000009.

A000110 counts set partitions.

A072233 counts partitions by sum and length.

Cf. A001055, A001970, A011782, A055887, A061260, A270995, A279784, A294617, A304969.

KEYWORD

easy,nonn,changed

AUTHOR

Vladeta Jovovic, Jun 11 2005

EXTENSIONS

More terms from Paul D. Hanna, Jun 26 2005

STATUS

approved

A050361

Number of factorizations into distinct prime powers greater than 1.

+10
22

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)

Index entries for sequences computed from exponents in factorization of n.

FORMULA

Dirichlet g.f.: Product_{n is a prime power >1}(1 + 1/n^s).

Multiplicative with a(p^e) = A000009(e).

a(A002110(k))=1.

a(n) = A050362(A101296(n)). - R. J. Mathar, May 26 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.26020571070524171076..., where f(x) = (1-x) * Product_{k>=1} (1 + x^k). - Amiram Eldar, Oct 03 2023

EXAMPLE

From Gus Wiseman, Jul 30 2022: (Start)

The A000688(216) = 9 factorizations of 216 into prime powers are:

(2*2*2*3*3*3)

(2*2*2*3*9)

(2*2*2*27)

(2*3*3*3*4)

(2*3*4*9)

(2*4*27)

(3*3*3*8)

(3*8*9)

(8*27)

Of these, the a(216) = 4 strict cases are:

(2*3*4*9)

(2*4*27)

(3*8*9)

(8*27)

(End)

MAPLE

A050361 := proc(n)

local a, f;

if n = 1 then

else

a := 1 ;

for f in ifactors(n)[2] do

a := a*A000009(op(2, f)) ;

end do:

end if;

end proc: # R. J. Mathar, May 25 2017

MATHEMATICA

Table[Times @@ PartitionsQ[Last /@ FactorInteger[n]], {n, 99}] (* Arkadiusz Wesolowski, Feb 27 2017 *)

PROG

(Haskell)

a050361 = product . map a000009 . a124010_row

-- Reinhard Zumkeller, Aug 28 2014

(PARI)

A000009(n, k=(n-!(n%2))) = if(!n, 1, my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k, k)); k -= 2); (s));

A050361(n) = factorback(apply(A000009, factor(n)[, 2])); \\ Antti Karttunen, Nov 17 2019

CROSSREFS

Cf. A000009, A050360, A050362, A050363, A050364, A101296.

Cf. A124010.

This is the strict case of A000688.

Positions of 1's are A004709, complement A046099.

The case of primes (instead of prime-powers) is A008966, non-strict A000012.

The non-strict additive version allowing 1's A023893, ranked by A302492.

The non-strict additive version is A023894, ranked by A355743.

The additive version (partitions) is A054685, ranked by A356065.

The additive version allowing 1's is A106244, ranked by A302496.

A001222 counts prime-power divisors.

A005117 lists all squarefree numbers.

A034699 gives maximal prime-power divisor.

A246655 lists all prime-powers (A000961 includes 1), towers A164336.

A296131 counts twice-factorizations of type PQR, non-strict A295935.

Cf. A001970, A002110, A025487, A055887, A063834, A076610, A085970, A279786, A302590, A302601, A354911, A355742.

KEYWORD

nonn,easy,mult

AUTHOR

Christian G. Bower, Oct 15 1999

STATUS

approved

A356065

Squarefree numbers whose prime indices are all prime-powers.

+10
10

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..58.

FORMULA

Intersection of A005117 and A355743.

EXAMPLE

105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]

CROSSREFS

The multiplicative version (factorizations) is A050361, non-strict A000688.

Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.

Counting twice-partitions of this type gives A279786, non-strict A279784.

Counting twice-factorizations gives A295935, non-strict A296131.

These are the odd products of distinct elements of A302493.

Allowing prime index 1 gives A302496, non-strict A302492.

The case of primes (instead of prime-powers) is A302590, non-strict A076610.

These are the squarefree positions of 1's in A355741.

This is the squarefree case of A355743, complement A356066.

A001222 counts prime-power divisors.

A005117 lists the squarefree numbers.

A034699 gives maximal prime-power divisor.

A246655 lists the prime-powers (A000961 includes 1), towers A164336.

A355742 chooses a prime-power divisor of each prime index.

Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 25 2022

STATUS

approved

A302496

Products of distinct primes of prime-power index.

+10
3

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 38, 41, 42, 46, 51, 53, 55, 57, 59, 62, 66, 67, 69, 70, 77, 82, 83, 85, 93, 95, 97, 102, 103, 105, 106, 109, 110, 114, 115, 118, 119, 123, 127, 131, 133, 134, 138, 154, 155, 157, 159

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A prime index of n is a number m such that prime(m) divides n.

LINKS

Table of n, a(n) for n=1..61.

EXAMPLE

Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of constant-multiset systems.

01: {}

02: {{}}

03: {{1}}

05: {{2}}

06: {{},{1}}

07: {{1,1}}

10: {{},{2}}

11: {{3}}

14: {{},{1,1}}

15: {{1},{2}}

17: {{4}}

19: {{1,1,1}}

21: {{1},{1,1}}

22: {{},{3}}

23: {{2,2}}

30: {{},{1},{2}}

31: {{5}}

33: {{1},{3}}

34: {{},{4}}

35: {{2},{1,1}}

38: {{},{1,1,1}}

MATHEMATICA

Select[Range[nn], Or[#===1, SquareFreeQ[#]&&And@@PrimePowerQ/@PrimePi/@DeleteCases[FactorInteger[#][[All, 1]], 2]]&]

PROG

(PARI) is(n) = if(bigomega(n)!=omega(n), return(0), my(f=factor(n)[, 1]~); for(k=1, #f, if(!isprimepower(primepi(f[k])) && primepi(f[k])!=1, return(0)))); 1 \\ Felix Fröhlich, Apr 10 2018

CROSSREFS

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279786, A281113, A296131, A301767, A302242, A302243, A302493, A302494.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 09 2018

STATUS

approved

A301767

Number of ways to choose a constant rooted partition of each part in a strict rooted partition of n.

+10
1

1, 1, 1, 3, 4, 7, 9, 15, 21, 32, 45, 59, 89, 117, 162, 225, 309, 394, 538, 707, 929, 1240, 1613, 2055, 2677, 3517, 4439, 5724, 7288, 9222, 11671, 14809, 18480, 23226, 29138, 36501, 45373, 56438, 69920, 86426, 106715, 131171, 161428, 197717, 242301, 295888

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

A rooted partition of n is an integer partition of n - 1.

LINKS

Table of n, a(n) for n=1..46.

FORMULA

O.g.f.: Product_{n>0} (1 + d(n-1) x^n) where d(n) = A000005(n) and d(0) = 1.

EXAMPLE

The a(7) = 9 rooted twice-partitions:

(5), (11111),

(4)(), (22)(), (1111)(), (3)(1), (111)(1),

(2)(1)(), (11)(1)().

MATHEMATICA

Table[Sum[Product[If[k===1, 1, DivisorSigma[0, k-1]], {k, ptn}], {ptn, Select[IntegerPartitions[n-1], UnsameQ@@#&]}], {n, 50}]

CROSSREFS

Cf. A002865, A032305, A063834, A093637, A279786, A296131, A301422, A301462, A301467, A301480, A301706.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 26 2018

STATUS

approved

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