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A294019 Number of same-trees whose leaves are the parts of the integer partition with Heinz number n. 3
0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 3, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

By convention a(1) = 0.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

A281145(n) = Sum_{i=1..A000041(n)} a(A215366(n,i)).

a(p^n) = A006241(n) for any prime p and exponent n >= 1. - Antti Karttunen, Sep 22 2018

EXAMPLE

The a(108) = 8 same-trees: ((22)(2(11))), ((22)((11)2)), ((2(11))(22)), (((11)2)(22)), (222(11)), (22(11)2), (2(11)22), ((11)222).

From Antti Karttunen, Sep 22 2018: (Start)

For 12 = prime(1)^2 * prime(2)^1, we have the following two cases: 2(11) and (11)2, thus a(12) = 2.

For 36 = prime(1)^2 * prime(2)^2, we have the following cases: (11)22, 2(11)2, 22(11), thus a(36) = 3.

For 144  = prime(1)^4 * prime(2)^2, we have the following 14 cases: (1111)(22), (22)(1111); ((11)(11))(22), (22)((11)(11)); (11)(11)22, (11)2(11)2, (11)22(11), 2(11)2(11), 2(11)(11)2, 22(11)(11); ((11)2)(11(2)), ((11)2)(2(11)), (2(11))((11)2), (2(11))(2(11)), thus a(144) = 14.

For n = 8775 = 3^3 * 5^2 * 13^1 = prime(2)^3 * prime(3)^2 * prime(6)^1, we have the following six cases: (222)(33)6, (222)6(33), (33)(222)6, (33)6(222), 6(222)(33), 6(33)(222), thus a(8775) = 6.

(End)

MATHEMATICA

nn=120;

ptns=Table[If[n===1, {}, Join@@Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]], {n, nn}];

tris=Join@@Map[Tuples[IntegerPartitions/@#]&, ptns];

qci[y_]:=qci[y]=If[Length[y]===1, 1, Sum[Times@@qci/@t, {t, Select[tris, And[Length[#]>1, Sort[Join@@#, Greater]===y, SameQ@@Total/@#]&]}]];

qci/@ptns

PROG

(PARI)

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }

productifbalancedfactorization(v) = if(!#v, 1, my(pw=A056239(v[1]), m=1); for(i=1, #v, if(A056239(v[i])!=pw, return(0), m *= A294019(v[i]))); (m));

A294019aux(n, m, facs) = if(1==n, productifbalancedfactorization(Vec(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A294019aux(n/d, m, newfacs))); (s));

A294019(n) = if(1==n, 0, if(isprime(n), 1, A294019aux(n, n-1, List([]))));

\\ A memoized implementation:

map294019 = Map();

A294019(n) = if(1==n, 0, if(isprime(n), 1, if(mapisdefined(map294019, n), mapget(map294019, n), my(v=A294019aux(n, n-1, List([]))); mapput(map294019, n, v); (v)))); \\ Antti Karttunen, Sep 22 2018

CROSSREFS

Cf. A000005, A000041, A000720, A001222, A006241, A056239, A063834, A196545, A215366, A273873, A281145, A289501, A296150, A299200, A299201, A299202, A299203, A294018, A294080.

Sequence in context: A280586 A112344 A294080 * A123721 A077618 A085863

Adjacent sequences:  A294016 A294017 A294018 * A294020 A294021 A294022

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 07 2018

STATUS

approved

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)