A329747 Runs-resistance of the sequence of prime indices of n.

0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 3, 0, 3, 2, 2, 0, 3, 1, 2, 1, 3, 0, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 3, 0, 2, 0, 3, 3, 2, 0, 3, 1, 3, 2, 3, 0, 3, 2, 3, 2, 2, 0, 4, 0, 2, 3, 1, 2, 2, 0, 3, 2, 2, 0, 3, 0, 2, 3, 3, 2, 2, 0, 3, 1, 2, 0, 4, 2, 2, 2, 3, 0, 3, 2, 3, 2, 2, 2, 3, 0, 3, 3, 2, 0, 2, 0, 3, 2, 2, 0, 3, 0, 2, 2, 3, 0, 2, 2, 3, 3, 2, 2, 4

Offset

1,6

Comments

First differs from A304455 at a(90) = 3, A304455(90) = 4.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

A prime index of n is a number m such that prime(m) divides n. The sequence of prime indices of n is row n of A112798.

Links

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

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Index entries for sequences related to prime indices in the factorization of n.

Example

We have (1,2,2,3) -> (1,2,1) -> (1,1,1) -> (3), so a(90) = 3.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
Table[runsres[primeMS[n]], {n, 50}]

Prog

(PARI)
pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1, #f~, while(f[i, 2], listput(runs, primepi(f[i, 1])); f[i, 2]--)); (runs); };
runlengths(lista) = if(!#lista, lista, if(1==#lista, List([1]), my(runs=List([]), rl=1); for(i=1, #lista, if((i< #lista) && (lista[i]==lista[i+1]), rl++, listput(runs, rl); rl=1)); (runs)));
A329747(n) = { my(runs=pis_to_runs(n)); for(i=0, oo, if(#runs<=1, return(i), runs = runlengths(runs))); }; \\ Antti Karttunen, Jan 20 2025

Crossrefs

The version for partitions is A329746.

The version for compositions is A329744.

The version for binary words is A329767.

The version for binary expansion is A318928.

Cf. A008578 (positions of 0's), A056239, A112798, A329745, A329750.

Sequence in context: A337541 A185279 A088432 * A304455 A345354 A238951

Adjacent sequences: A329744 A329745 A329746 * A329748 A329749 A329750

Keyword

nonn

Author

Gus Wiseman, Nov 21 2019

Extensions

More terms from Antti Karttunen, Jan 20 2025

Status

approved