A325355 One plus the number of steps applying A325351 (Heinz number of augmented differences of reversed prime indices) to reach a fixed point.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 5, 1, 4, 2, 1, 1, 6, 1, 1, 4, 1, 1, 3, 1, 1, 1, 2, 2, 7, 1, 1, 2, 3, 1, 8, 1, 1, 3, 1, 1, 4, 1, 5, 5, 1, 1, 9, 4, 1, 2, 1, 1, 3, 1, 5, 6, 1, 1, 2, 1, 1, 4, 4, 1, 10, 1, 1, 3, 5, 1, 11, 1, 6, 1, 1, 2, 5, 2, 1, 7, 1, 1, 3

Offset

1,9

Comments

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

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The fixed points of A325351 are the Heinz numbers of hooks A093641.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to Heinz numbers

Example

Repeatedly applying A325351 starting with 78 gives 78 -> 66 -> 42 -> 30 -> 18 -> 12, and 12 is a fixed point, so a(78) = 6.

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[FixedPointList[Times@@Prime/@aug[primeptn[#]]&, n]]-1, {n, 50}]

Prog

(PARI)
augdiffs(n) = { my(diffs=List([]), f=factor(n), prevpi, pi=0, i=#f~); while(i, prevpi=pi; pi = primepi(f[i, 1]); if(prevpi, listput(diffs, 1+(prevpi-pi))); if(f[i, 2]>1, f[i, 2]--, i--)); if(pi, listput(diffs, pi)); Vec(diffs); };
A325351(n) = factorback(apply(prime, augdiffs(n)));
A325355(n) = { my(u=A325351(n)); if(u==n, 1, 1+A325355(u)); }; \\ Antti Karttunen, Nov 16 2019

Crossrefs

Positions of 2's are A325359.

Cf. A056239, A093641, A112798, A130091, A289509, A325351, A325352, A325366, A325389, A325394, A325395, A325396.

Sequence in context: A328672 A368465 A378222 * A219093 A062760 A323163

Adjacent sequences: A325352 A325353 A325354 * A325356 A325357 A325358

Keyword

nonn

Author

Gus Wiseman, Apr 23 2019

Extensions

More terms from Antti Karttunen, Nov 16 2019

Status

approved