A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

1, 9, 25, 27, 121, 169, 289, 81, 125, 841, 961, 675, 1681, 1849, 2209, 243, 3481, 1125, 4489, 3267, 5329, 6241, 6889, 2025, 1331, 10201, 625, 7803, 11881, 12769, 16129, 729, 18769, 19321, 22201, 2197, 24649, 26569, 27889, 9801, 32041, 32761, 36481, 25947

Offset: 1

Author

Gus Wiseman, Feb 14 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 first position of 12 in A290106 is 675, with prime indices {2,2,2,3,3}, so a(12) = 675.
The terms together with their prime indices begin:
      1: {}
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    169: {6,6}
    289: {7,7}
     81: {2,2,2,2}
    125: {3,3,3}
    841: {10,10}
    961: {11,11}
    675: {2,2,2,3,3}
   1681: {13,13}
   1849: {14,14}
   2209: {15,15}
    243: {2,2,2,2,2}
   3481: {17,17}
   1125: {2,2,3,3,3}
		

Crossrefs

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

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

Position of first appearance of n in A290106.

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

For difference instead of quotient see A380986.

The sorted version is A380988.

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}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,10000}];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]