A325133 Heinz number of the integer partition obtained by removing the inner lining, or, equivalently, the largest hook, of the integer partition with Heinz number n.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 5, 3, 2, 1, 1, 4, 3, 1, 2, 1, 1, 2, 1, 1, 4, 1, 3, 2, 1, 1, 2, 3, 1, 2, 1, 1, 6, 1, 5, 2, 1, 1, 8, 1, 1, 2, 3, 1, 2, 1, 1, 4, 5, 1, 2, 1, 3, 1, 1, 5, 4, 3, 1, 2, 1, 1, 6

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 is the partition (4,2) with Heinz number 21, so a(715) = 21.
		

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 ones are A093641 (Heinz numbers of hooks). The number of iterations required to reach 1 starting with n is A257990(n).

Cf. A000720, A001222, A046660, A052126, A056239, A061395, A064989, A112798, A243055, A252464, A325134, A325135.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,1,Times@@Prime/@DeleteCases[Most[primeMS[n]]-1,0]],{n,100}]

PARI

A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
A325133(n) = A052126(A064989(n)); \\ Antti Karttunen, Apr 14 2019

Formula

a(n) = A064989(A052126(n)) = A052126(A064989(n)).

Extensions

More terms from Antti Karttunen, Apr 14 2019