A358526 Numbers k whose prime indices have a different number of permutations than any number less than k.

1, 6, 12, 24, 30, 48, 60, 72, 120, 144, 180, 192, 210, 288, 360, 384, 432, 480, 576, 720, 768, 840, 864, 900, 1080, 1152, 1260, 1296, 1440, 1680, 1728, 1920, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 4320, 4608, 4620, 5040, 5400, 6300, 6912, 7200, 7560, 7680

Offset: 1

Author

Gus Wiseman, Nov 21 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.

Permutations of prime indices are counted by A008480.

Examples

			The terms together with their prime indices begin:
      1: {}
      6: {1,2}
     12: {1,1,2}
     24: {1,1,1,2}
     30: {1,2,3}
     48: {1,1,1,1,2}
     60: {1,1,2,3}
     72: {1,1,1,2,2}
    120: {1,1,1,2,3}
    144: {1,1,1,1,2,2}
    180: {1,1,2,2,3}
    192: {1,1,1,1,1,1,2}
    210: {1,2,3,4}
    288: {1,1,1,1,1,2,2}
    360: {1,1,1,2,2,3}
    384: {1,1,1,1,1,1,1,2}
    432: {1,1,1,1,2,2,2}
    480: {1,1,1,1,1,2,3}
		

Crossrefs

Sorted list of positions of first appearances in A008480.

The unsorted version is A304938.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A289509 lists numbers with relatively prime prime indices.

A344606 counts alternating permutations of prime indices.

Cf. A000040, A000720, A001221, A003963, A061395, A316524.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fir[q_]:=Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&];
fir[Table[Length[Permutations[primeMS[n]]],{n,1000}]]