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A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

%I A355527 #9 Jul 13 2022 20:37:00
%S A355527 6,15,30,35,42,66,70,77,78,102,105,114,138,143,154,165,174,186,195,
%T A355527 210,221,222,231,246,255,258,282,285,286,318,323,330,345,354,366,385,
%U A355527 390,402,426,429,435,437,438,442,455,462,465,474,498,510,534,546,555,570
%N A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.
%C A355527 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 A355527 A number is squarefree if it is not divisible by any perfect square > 1.
%C A355527 A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.
%H A355527 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%F A355527 Intersection of A005117 (squarefree) and A104210 (has consecutive primes).
%e A355527 The terms together with their prime indices begin:
%e A355527     6: {1,2}
%e A355527    15: {2,3}
%e A355527    30: {1,2,3}
%e A355527    35: {3,4}
%e A355527    42: {1,2,4}
%e A355527    66: {1,2,5}
%e A355527    70: {1,3,4}
%e A355527    77: {4,5}
%e A355527    78: {1,2,6}
%e A355527   102: {1,2,7}
%e A355527   105: {2,3,4}
%e A355527   114: {1,2,8}
%e A355527   138: {1,2,9}
%e A355527   143: {5,6}
%e A355527   154: {1,4,5}
%e A355527   165: {2,3,5}
%e A355527   174: {1,2,10}
%e A355527   186: {1,2,11}
%e A355527   195: {2,3,6}
%e A355527   210: {1,2,3,4}
%t A355527 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A355527 Select[Range[100],Min@@Differences[primeMS[#]]==1&]
%Y A355527 Crossrefs found in the link are not repeated here.
%Y A355527 All terms are in A005117, complement A013929.
%Y A355527 For minimal difference <= 1 we have A055932.
%Y A355527 For maximal instead of minimal difference = 1 we have A066312.
%Y A355527 For minimal difference > 1 we have A325160.
%Y A355527 If zero is considered a prime index we get A355530.
%Y A355527 A001522 counts partitions with a fixed point (unproved), ranked by A352827.
%Y A355527 A287352, A355533, A355534, A355536 list the differences of prime indices.
%Y A355527 A355524 or A355525 give minimal difference between prime indices.
%Y A355527 Cf. A000005, A000040, A056239, A120944, A130091, A238353, A238354, A286470, A325161, A352822, A355526, A355531.
%K A355527 nonn
%O A355527 1,1
%A A355527 _Gus Wiseman_, Jul 10 2022