A355530 - OEIS

%I #9 Jul 13 2022 20:37:13

%S 2,6,10,14,15,22,26,30,34,35,38,42,46,58,62,66,70,74,77,78,82,86,94,

%T 102,105,106,110,114,118,122,130,134,138,142,143,146,154,158,165,166,

%U 170,174,178,182,186,190,194,195,202,206,210,214,218,221,222,226,230

%N Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

%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.

%C A number is squarefree if it is not divisible by any perfect square > 1.

%C A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

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

%F Equals A005117 /\ (A005843 \/ A104210).

%e The terms together with their prime indices begin:

%e    2: {1}

%e    6: {1,2}

%e   10: {1,3}

%e   14: {1,4}

%e   15: {2,3}

%e   22: {1,5}

%e   26: {1,6}

%e   30: {1,2,3}

%e   34: {1,7}

%e   35: {3,4}

%e   38: {1,8}

%e   42: {1,2,4}

%e   46: {1,9}

%e   58: {1,10}

%e   62: {1,11}

%e   66: {1,2,5}

%e   70: {1,3,4}

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

%t Select[Range[100],Min@@Differences[Prepend[primeMS[#],0]]==1&]

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

%Y All terms are in A005117, complement A013929.

%Y For maximal instead of minimal difference we have A055932 or A066312.

%Y Not prepending zero gives A355527.

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

%Y A056239 adds up prime indices.

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

%Y A279945 counts partitions by number of distinct differences.

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

%Y A355524 gives minimal difference if singletons go to 0, to index A355525.

%Y Cf. A000005, A000040, A120944, A238354, A286469, A286470, A325160, A325161, A355526, A355531.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jul 10 2022