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

1,9

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

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

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}]

Prog

(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

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.

Sequence in context: A033272 A324907 A331295 * A236338 A284259 A250068

Adjacent sequences: A325130 A325131 A325132 * A325134 A325135 A325136

Keyword

nonn

Author

Gus Wiseman, Apr 02 2019

Extensions

More terms from Antti Karttunen, Apr 14 2019

Status

approved