A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

1, 9, 25, 27, 81, 121, 125, 169, 243, 289, 625, 675, 729, 841, 961, 1125, 1331, 1681, 1849, 2025, 2187, 2197, 2209, 3125, 3267, 3481, 4489, 4913, 5329, 5625, 6075, 6241, 6561, 6889, 7803, 9801, 10125, 10201, 11881, 11979, 12769, 14641, 15125, 15625, 16129

Offset: 1

Author

Gus Wiseman, Feb 18 2025

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.

All terms are odd.

Examples

			The prime indices of 225 are {2,2,3,3}, with image A290106(225) = 6. The prime indices of 169 are {6,6}, also with image 6. Since the latter is the first with image 6, 169 is in the sequence, and 225 is not.
The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    81: {2,2,2,2}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   243: {2,2,2,2,2}
   289: {7,7}
   625: {3,3,3,3}
   675: {2,2,2,3,3}
   729: {2,2,2,2,2,2}
   841: {10,10}
   961: {11,11}
  1125: {2,2,3,3,3}
  1331: {5,5,5}
  1681: {13,13}
  1849: {14,14}
  2025: {2,2,2,2,3,3}
		

Crossrefs

For factors instead of indices we have A001694 (unsorted A064549), firsts of A003557.

Sorted firsts of A290106.

The additive version is A380957 (sorted A380956), firsts of A380955.

For difference instead of quotient see A380986.

The unsorted version is A380987.

The additive version for factors is A381075 (unsorted A280286), firsts of A280292.

A000040 lists the primes, differences A001223.

A003963 gives product of prime indices, distinct A156061.

A005117 lists squarefree numbers, complement A013929.

A055396 gives least prime index, greatest A061395.

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

A304038 lists distinct prime indices, sum A066328, length A001221.

Cf. A000720, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]