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A326844
Let y be the integer partition with Heinz number n. Then a(n) is the size of the complement, in the minimal rectangular partition containing the Young diagram of y, of the Young diagram of y.
22
0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 3, 1, 0, 0, 1, 0, 4, 2, 4, 0, 3, 0, 5, 0, 6, 0, 3, 0, 0, 3, 6, 1, 2, 0, 7, 4, 6, 0, 5, 0, 8, 2, 8, 0, 4, 0, 2, 5, 10, 0, 1, 2, 9, 6, 9, 0, 5, 0, 10, 4, 0, 3, 7, 0, 12, 7, 4, 0, 3, 0, 11, 1, 14, 1, 9, 0, 8, 0, 12, 0, 8, 4, 13, 8, 12, 0, 4, 2, 16, 9, 14, 5, 5, 0, 3, 6, 4
OFFSET
1,10
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
LINKS
FORMULA
a(n) = A001222(n) * A061395(n) - A056239(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 . . .
The size of the complement (shown in dots) in a 6 X 4 rectangle is 5, so a(7865) = 5.
MATHEMATICA
Table[If[n==1, 0, With[{y=Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]}, Max[y]*Length[y]-Total[y]]], {n, 100}]
PROG
(PARI)
A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A326844(n) = ((bigomega(n)*A061395(n)) - A056239(n)); \\ Antti Karttunen, Feb 10 2023
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 26 2019
EXTENSIONS
Data section extended up to term a(100) by Antti Karttunen, Feb 10 2023
STATUS
approved