A325133 - OEIS

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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.

7

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

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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