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A355527
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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.
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5
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6, 15, 30, 35, 42, 66, 70, 77, 78, 102, 105, 114, 138, 143, 154, 165, 174, 186, 195, 210, 221, 222, 231, 246, 255, 258, 282, 285, 286, 318, 323, 330, 345, 354, 366, 385, 390, 402, 426, 429, 435, 437, 438, 442, 455, 462, 465, 474, 498, 510, 534, 546, 555, 570
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OFFSET
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1,1
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COMMENTS
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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.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.
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LINKS
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FORMULA
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Intersection of A005117 (squarefree) and A104210 (has consecutive primes).
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EXAMPLE
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The terms together with their prime indices begin:
6: {1,2}
15: {2,3}
30: {1,2,3}
35: {3,4}
42: {1,2,4}
66: {1,2,5}
70: {1,3,4}
77: {4,5}
78: {1,2,6}
102: {1,2,7}
105: {2,3,4}
114: {1,2,8}
138: {1,2,9}
143: {5,6}
154: {1,4,5}
165: {2,3,5}
174: {1,2,10}
186: {1,2,11}
195: {2,3,6}
210: {1,2,3,4}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Min@@Differences[primeMS[#]]==1&]
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CROSSREFS
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Crossrefs found in the link are not repeated here.
For minimal difference <= 1 we have A055932.
For maximal instead of minimal difference = 1 we have A066312.
For minimal difference > 1 we have A325160.
If zero is considered a prime index we get A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
Cf. A000005, A000040, A056239, A120944, A130091, A238353, A238354, A286470, A325161, A352822, A355526, A355531.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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