cp's OEIS Frontend

A355523 Number of distinct differences between adjacent prime indices of n.

%I A355523 #18 Jan 20 2025 10:21:28
%S A355523 0,0,0,1,0,1,0,1,1,1,0,2,0,1,1,1,0,2,0,2,1,1,0,2,1,1,1,2,0,1,0,1,1,1,
%T A355523 1,2,0,1,1,2,0,2,0,2,2,1,0,2,1,2,1,2,0,2,1,2,1,1,0,2,0,1,2,1,1,2,0,2,
%U A355523 1,2,0,2,0,1,2,2,1,2,0,2,1,1,0,3,1,1,1,2,0,2,1,2,1,1,1,2,0,2,2,2,0,2,0,2,1
%N A355523 Number of distinct differences between adjacent prime indices of n.
%C A355523 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%H A355523 Antti Karttunen, <a href="/A355523/b355523.txt">Table of n, a(n) for n = 1..65537</a>
%H A355523 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%H A355523 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A355523 For example, the prime indices of 22770 are {1,2,2,3,5,9}, with differences (1,0,1,2,4), so a(22770) = 4.
%t A355523 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355523 Table[Length[Union[Differences[primeMS[n]]]],{n,1000}]
%o A355523 (PARI) A355523(n) = if(1==n, 0, my(pis = apply(primepi,factor(n)[,1]), difs = vector(#pis-1, i, pis[i+1]-pis[i])); (#Set(difs)+!issquarefree(n))); \\ _Antti Karttunen_, Jan 20 2025
%Y A355523 Crossrefs found in the link are not repeated here.
%Y A355523 Counting m such that A056239(m) = n and a(m) = k gives A279945.
%Y A355523 With multiplicity we have A252736(n) = A001222(n) - 1.
%Y A355523 The maximal difference is A286470, minimal A355524.
%Y A355523 A008578 gives the positions of 0's.
%Y A355523 A287352 lists differences between 0-prepended prime indices.
%Y A355523 A355534 lists augmented differences between prime indices.
%Y A355523 A355536 lists differences between prime indices.
%Y A355523 Cf. A066312, A238353, A238710, A286469, A320348, A325388, A325406, A351294, A352827, A355525-A355533.
%K A355523 nonn
%O A355523 1,12
%A A355523 _Gus Wiseman_, Jul 10 2022
%E A355523 Data section extended to a(105) by _Antti Karttunen_, Jan 20 2025