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A300385
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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.
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4
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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
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OFFSET
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1,12
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COMMENTS
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The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300384.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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