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A325166
Size of the internal portion of the integer partition with Heinz number n.
14
0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 3, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 3, 0, 1, 3, 1, 0, 1, 3, 3, 2, 1, 0, 3, 3, 1, 2, 1, 0, 3, 0, 1, 3, 0, 3, 3, 0, 1, 2, 4, 0, 2, 0, 1, 4, 1, 4, 3, 0, 1, 3, 1, 0, 3, 3, 1, 2, 1, 0, 4, 4, 1, 2, 1, 3, 1, 0, 4, 3, 3, 0, 3, 0, 1, 5
OFFSET
1,15
COMMENTS
The internal portion of an integer partition consists of all squares in the Young diagram that have a square both directly below and directly to the right.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
FORMULA
a(n) = A056239(n) - A061395(n) - A001222(n) + A001221(n).
a(n) = A056239(n) - A297113(n).
EXAMPLE
The partition with Heinz number 7865 is (6,5,5,3), with diagram
o o o o o o
o o o o o
o o o o o
o o o
with internal portion
o o o o o
o o o o
o o o
of size 12, so a(7865) = 12.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[n==1, 0, Total[primeMS[n]]-Max[primeMS[n]]-Length[primeMS[n]]+Length[Union[primeMS[n]]]], {n, 100}]
PROG
(PARI)
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A325166(n) = (A056239(n) - A061395(n) - bigomega(n) + omega(n)); \\ Antti Karttunen, Apr 14 2019
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 05 2019
EXTENSIONS
More terms from Antti Karttunen, Apr 14 2019
STATUS
approved