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

Author

Gus Wiseman, Apr 05 2019

Keywords

• nonn

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).

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Positions of zeros are A174090.

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

Programs

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}]

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

Formula

a(n) = A056239(n) - A061395(n) - A001222(n) + A001221(n).

a(n) = A056239(n) - A297113(n).

Extensions

More terms from Antti Karttunen, Apr 14 2019