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

1,25

Comments

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(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.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A092078 A360071 A358724 * A263577 A343631 A342561

Adjacent sequences: A325132 A325133 A325134 * A325136 A325137 A325138

Keyword

nonn

Author

Gus Wiseman, Apr 02 2019

Extensions

More terms from Antti Karttunen, Apr 14 2019

Status

approved