A355525 - OEIS

%I #8 Jul 11 2022 08:33:43

%S 1,2,0,3,1,4,0,0,2,5,0,6,3,1,0,7,0,8,0,2,4,9,0,0,5,0,0,10,1,11,0,3,6,

%T 1,0,12,7,4,0,13,1,14,0,0,8,15,0,0,0,5,0,16,0,2,0,6,9,17,0,18,10,0,0,

%U 3,1,19,0,7,1,20,0,21,11,0,0,1,1,22,0,0,12

%N Minimal difference between adjacent prime indices of n, or k if n is the k-th prime.

%C 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 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[If[PrimeQ[n],PrimePi[n],Min@@Differences[primeMS[n]]],{n,2,100}]

%Y Crossrefs found in the link are not repeated here.

%Y Positions of first appearances are 4 followed by A000040.

%Y Positions of 0's are A013929, see also A130091.

%Y Triangle A238709 counts m such that A056239(m) = n and a(m) = k.

%Y For maximal instead of minimal difference we have A286470.

%Y Positions of terms > 1 are A325160, also A325161.

%Y See also A355524, A355528.

%Y Positions of 1's are A355527.

%Y A001522 counts partitions with a fixed point (unproved), ranked by A352827.

%Y A238352 counts partitions by fixed points, rank statistic A352822.

%Y A287352, A355533, A355534, A355536 list the differences of prime indices.

%Y Cf. A066312, A120944, A238353, A279945, A286469, A325351, A325352, A351294, A355526, A355530, A355531, A355532.

%K nonn

%O 2,2

%A _Gus Wiseman_, Jul 10 2022