A325166 Size of the internal portion of the integer partition with Heinz number n.

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

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

Crossrefs

Positions of zeros are A174090.

Cf. A001221, A001222, A052126, A056239, A061395, A064989, A065770, A112798, A252464, A257990, A297113, A325133, A325135, A325167, A325169.

Sequence in context: A262726 A112605 A111775 * A025844 A035461 A118508

Adjacent sequences: A325163 A325164 A325165 * A325167 A325168 A325169

Keyword

nonn

Author

Gus Wiseman, Apr 05 2019

Extensions

More terms from Antti Karttunen, Apr 14 2019

Status

approved