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.

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

Offset: 1

Author

Gus Wiseman, Jul 10 2022

Keywords

• nonn

Comments

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.

Examples

			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}
		

Links

• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

Crossrefs found in the link are not repeated here.

All terms are in A005117, complement A013929.

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.

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

A355524 or A355525 give minimal difference between prime indices.

Cf. A000005, A000040, A056239, A120944, A130091, A238353, A238354, A286470, A325161, A352822, A355526, A355531.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Min@@Differences[primeMS[#]]==1&]

Formula

Intersection of A005117 (squarefree) and A104210 (has consecutive primes).