cp's OEIS Frontend

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

%I A380988 #5 Feb 18 2025 15:34:06
%S A380988 1,9,25,27,81,121,125,169,243,289,625,675,729,841,961,1125,1331,1681,
%T A380988 1849,2025,2187,2197,2209,3125,3267,3481,4489,4913,5329,5625,6075,
%U A380988 6241,6561,6889,7803,9801,10125,10201,11881,11979,12769,14641,15125,15625,16129
%N A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).
%C A380988 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.
%C A380988 All terms are odd.
%e A380988 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.
%e A380988 The terms together with their prime indices begin:
%e A380988      1: {}
%e A380988      9: {2,2}
%e A380988     25: {3,3}
%e A380988     27: {2,2,2}
%e A380988     81: {2,2,2,2}
%e A380988    121: {5,5}
%e A380988    125: {3,3,3}
%e A380988    169: {6,6}
%e A380988    243: {2,2,2,2,2}
%e A380988    289: {7,7}
%e A380988    625: {3,3,3,3}
%e A380988    675: {2,2,2,3,3}
%e A380988    729: {2,2,2,2,2,2}
%e A380988    841: {10,10}
%e A380988    961: {11,11}
%e A380988   1125: {2,2,3,3,3}
%e A380988   1331: {5,5,5}
%e A380988   1681: {13,13}
%e A380988   1849: {14,14}
%e A380988   2025: {2,2,2,2,3,3}
%t A380988 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A380988 q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,1000}];
%t A380988 Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]
%Y A380988 For factors instead of indices we have A001694 (unsorted A064549), firsts of A003557.
%Y A380988 Sorted firsts of A290106.
%Y A380988 The additive version is A380957 (sorted A380956), firsts of A380955.
%Y A380988 For difference instead of quotient see A380986.
%Y A380988 The unsorted version is A380987.
%Y A380988 The additive version for factors is A381075 (unsorted A280286), firsts of A280292.
%Y A380988 A000040 lists the primes, differences A001223.
%Y A380988 A003963 gives product of prime indices, distinct A156061.
%Y A380988 A005117 lists squarefree numbers, complement A013929.
%Y A380988 A055396 gives least prime index, greatest A061395.
%Y A380988 A056239 adds up prime indices, row sums of A112798, length A001222.
%Y A380988 A304038 lists distinct prime indices, sum A066328, length A001221.
%Y A380988 Cf. A000720, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.
%K A380988 nonn
%O A380988 1,2
%A A380988 _Gus Wiseman_, Feb 18 2025