A300385 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the partition with Heinz number n to the local maximum.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 6, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 11, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 2, 11, 1, 3, 1, 2, 1, 3, 1, 19, 1, 1, 2, 2, 1, 3, 1, 14, 2, 1, 1, 10, 1, 1, 1, 5, 1, 10, 1, 2, 1, 1, 1, 33, 1, 2, 2, 7, 1, 3, 1, 5, 3

Offset

1,12

Comments

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..12288

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to Heinz numbers

Formula

a(1) = 0; for n > 1, if A001222(n) <= 2 [when n is a prime or semiprime], a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)+primepi(q)) * (n/(p*q))), where p ranges over all distinct primes dividing n, and q also ranges over primes dividing n, but with condition that q > p if p is a unitary prime factor of n, otherwise q >= p. Here primepi = A000720. - Antti Karttunen, Oct 07 2018

Example

The a(36) = 6 maximal chains are the rows:
(2211)<(222)<(42)<(6)
(2211)<(411)<(42)<(6)
(2211)<(411)<(51)<(6)
(2211)<(321)<(42)<(6)
(2211)<(321)<(51)<(6)
(2211)<(321)<(33)<(6)

Mathematica

chc[ptn_]:=If[Length[ptn]===1, 1, Total[chc/@Union[ReplaceList[ptn, {a___, x_, b___, y_, c___}:>Sort[{x+y, a, b, c}, Greater]]]]];
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[chc[Reverse[primeMS[n]]], {n, 100}]

Prog

(PARI) A300385(n) = if(1==n, 0, if(bigomega(n)<=2, 1, my(f=factor(n), u = #f~, s = 0); for(i=1, u, for(j=i+(1==f[i, 2]), u, s += A300385((n/(f[i, 1]*f[j, 1])*prime(primepi(f[i, 1])+primepi(f[j, 1])))))); (s))); \\ Antti Karttunen, Oct 06 2018
(PARI)
memoA300385 = Map();
A300385(n) = if(1==n, 0, if(bigomega(n)<=2, 1, if(mapisdefined(memoA300385, n), mapget(memoA300385, n), my(f=factor(n), u = #f~, s = 0); for(i=1, u, for(j=i+(1==f[i, 2]), u, s += A300385(prime(primepi(f[i, 1])+primepi(f[j, 1]))*(n/(f[i, 1]*f[j, 1]))))); mapput(memoA300385, n, s); (s)))); \\ (A memoized implementation). - Antti Karttunen, Oct 07 2018

Crossrefs

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300384.

Sequence in context: A344770 A293895 A375120 * A317145 A330665 A345985

Adjacent sequences: A300382 A300383 A300384 * A300386 A300387 A300388

Keyword

nonn

Author

Gus Wiseman, Mar 04 2018

Extensions

More terms from Antti Karttunen, Oct 06 2018

Status

approved