A355530 - OEIS

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.

2, 6, 10, 14, 15, 22, 26, 30, 34, 35, 38, 42, 46, 58, 62, 66, 70, 74, 77, 78, 82, 86, 94, 102, 105, 106, 110, 114, 118, 122, 130, 134, 138, 142, 143, 146, 154, 158, 165, 166, 170, 174, 178, 182, 186, 190, 194, 195, 202, 206, 210, 214, 218, 221, 222, 226, 230

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..57.

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

FORMULA

Equals A005117 /\ (A005843 \/ A104210).

EXAMPLE

The terms together with their prime indices begin:

2: {1}

6: {1,2}

10: {1,3}

14: {1,4}

15: {2,3}

22: {1,5}

26: {1,6}

30: {1,2,3}

34: {1,7}

35: {3,4}

38: {1,8}

42: {1,2,4}

46: {1,9}

58: {1,10}

62: {1,11}

66: {1,2,5}

70: {1,3,4}

MATHEMATICA

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

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

CROSSREFS

Crossrefs found in the link are not repeated here.

All terms are in A005117, complement A013929.

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

Not prepending zero gives A355527.

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

A056239 adds up prime indices.

A238352 counts partitions by fixed points, rank statistic A352822.

A279945 counts partitions by number of distinct differences.

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

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

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

Sequence in context: A165562 A302796 A302797 * A130319 A336338 A191256

Adjacent sequences: A355527 A355528 A355529 * A355531 A355532 A355533

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 10 2022

STATUS

approved