cp's OEIS Frontend

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

%I A325355 #13 Nov 16 2019 20:05:26
%S A325355 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,
%T A325355 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,
%U A325355 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
%N A325355 One plus the number of steps applying A325351 (Heinz number of augmented differences of reversed prime indices) to reach a fixed point.
%C A325355 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325355 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).
%C A325355 The fixed points of A325351 are the Heinz numbers of hooks A093641.
%H A325355 Antti Karttunen, <a href="/A325355/b325355.txt">Table of n, a(n) for n = 1..65537</a>
%H A325355 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A325355 <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%e A325355 Repeatedly applying A325351 starting with 78 gives 78 -> 66 -> 42 -> 30 -> 18 -> 12, and 12 is a fixed point, so a(78) = 6.
%t A325355 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325355 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325355 Table[Length[FixedPointList[Times@@Prime/@aug[primeptn[#]]&,n]]-1,{n,50}]
%o A325355 (PARI)
%o A325355 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); };
%o A325355 A325351(n) = factorback(apply(prime,augdiffs(n)));
%o A325355 A325355(n) = { my(u=A325351(n)); if(u==n,1,1+A325355(u)); }; \\ _Antti Karttunen_, Nov 16 2019
%Y A325355 Positions of 2's are A325359.
%Y A325355 Cf. A056239, A093641, A112798, A130091, A289509, A325351, A325352, A325366, A325389, A325394, A325395, A325396.
%K A325355 nonn
%O A325355 1,9
%A A325355 _Gus Wiseman_, Apr 23 2019
%E A325355 More terms from _Antti Karttunen_, Nov 16 2019