A325135 Size of the integer partition with Heinz number n after its inner lining, or, equivalently, its largest hook, is removed.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 2, 1, 0, 0, 2, 2, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 1, 0, 0, 1, 2, 0, 1, 0, 0, 3, 0, 3, 1, 0, 0, 3, 0, 0, 1, 2, 0, 1, 0, 0, 2, 3, 0, 1, 0, 2, 0, 0, 3, 2, 2, 0, 1, 0, 0, 3

Offset: 1

Author

Gus Wiseman, Apr 02 2019

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The partition with Heinz number 715 is (6,5,3), with diagram
  o o o o o o
  o o o o o
  o o o
which has inner lining
          o o
      o o o
  o o o
or largest hook
  o o o o o o
  o
  o
both of which have complement
  o o o o
  o o
which has size 6, so a(715) = 6.
		

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

Cf. A000720, A001222, A052126, A056239, A061395, A093641, A112798, A252464, A257990, A325133, A325134.

Programs

Mathematica

Table[If[n==1,0,Total[Most[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]-1]],{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])));
A325135(n) = if(1==n,0,(1+A056239(n)-bigomega(n)-A061395(n))); \\ Antti Karttunen, Apr 14 2019

Formula

a(n) = A056239(A325133(n)).

For n > 1:

a(n) = A056239(n) - A001222(n) - A061395(n) + 1.

a(n) = A056239(n) - A252464(n).

a(n) = A056239(n) - A325134(n) + 1.

Extensions

More terms from Antti Karttunen, Apr 14 2019