cp's OEIS Frontend

A355530 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.

%I A355530 #9 Jul 13 2022 20:37:13
%S A355530 2,6,10,14,15,22,26,30,34,35,38,42,46,58,62,66,70,74,77,78,82,86,94,
%T A355530 102,105,106,110,114,118,122,130,134,138,142,143,146,154,158,165,166,
%U A355530 170,174,178,182,186,190,194,195,202,206,210,214,218,221,222,226,230
%N A355530 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 A355530 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 A355530 A number is squarefree if it is not divisible by any perfect square > 1.
%C A355530 A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.
%H A355530 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%F A355530 Equals A005117 /\ (A005843 \/ A104210).
%e A355530 The terms together with their prime indices begin:
%e A355530    2: {1}
%e A355530    6: {1,2}
%e A355530   10: {1,3}
%e A355530   14: {1,4}
%e A355530   15: {2,3}
%e A355530   22: {1,5}
%e A355530   26: {1,6}
%e A355530   30: {1,2,3}
%e A355530   34: {1,7}
%e A355530   35: {3,4}
%e A355530   38: {1,8}
%e A355530   42: {1,2,4}
%e A355530   46: {1,9}
%e A355530   58: {1,10}
%e A355530   62: {1,11}
%e A355530   66: {1,2,5}
%e A355530   70: {1,3,4}
%t A355530 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355530 Select[Range[100],Min@@Differences[Prepend[primeMS[#],0]]==1&]
%Y A355530 Crossrefs found in the link are not repeated here.
%Y A355530 All terms are in A005117, complement A013929.
%Y A355530 For maximal instead of minimal difference we have A055932 or A066312.
%Y A355530 Not prepending zero gives A355527.
%Y A355530 A001522 counts partitions with a fixed point (unproved), ranked by A352827.
%Y A355530 A056239 adds up prime indices.
%Y A355530 A238352 counts partitions by fixed points, rank statistic A352822.
%Y A355530 A279945 counts partitions by number of distinct differences.
%Y A355530 A287352, A355533, A355534, A355536 list the differences of prime indices.
%Y A355530 A355524 gives minimal difference if singletons go to 0, to index A355525.
%Y A355530 Cf. A000005, A000040, A120944, A238354, A286469, A286470, A325160, A325161, A355526, A355531.
%K A355530 nonn
%O A355530 1,1
%A A355530 _Gus Wiseman_, Jul 10 2022