A331598 - OEIS

%I #15 Jan 31 2020 20:25:46

%S 1,1,1,1,1,2,1,1,3,4,1,2,1,8,1,1,1,1,1,4,2,16,1,2,9,32,5,8,1,2,1,1,4,

%T 64,3,3,1,128,8,4,1,4,1,16,1,256,1,2,27,1,16,32,1,5,1,8,32,512,1,6,1,

%U 1024,2,1,2,8,1,64,64,2,1,3,1,2048,5,128,9,16,1,4,7,4096,1,12,4,8192,128,16,1,10,3,256,256,16384,8,2,1,1,4,9,1,32,1,32,1

%N a(n) = A122111(n) / gcd(A122111(n),A241909(n)).

%C It appears that these and the terms of A331599 have the same prime signatures, that is, A046523(a(n)) = A046523(A331599(n)) seems to hold for all n. However, the sequences are not equivalence-class-wise same: a(6) = a(12) = 2, whereas A331599(6) = 3 and A331599(12) = 5.

%H Antti Karttunen, <a href="/A331598/b331598.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = A122111(n)/A331598(n) = A122111(n) / gcd(A122111(n),A241909(n)).

%F a(n) = A331599(A241916(n)).

%t Array[If[# == 1, 1, #1/GCD[#1, #2] & @@ {Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ #], Function[t, Times @@ Prime@ Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, #]]]]@ ConstantArray[0, Transpose[#][[1, -1]]] &[# /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]} &@ FactorInteger[#]] &, 90] (* _Michael De Vlieger_, Jan 25 2020, after _JungHwan Min_ at A122111 *)

%o (PARI) A331598(n) = { my(u=A122111(n)); u/gcd(u, A241909(n)); };

%Y Cf. A122111, A241909, A241916, A331594, A331595, A331599.

%K nonn

%O 1,6

%A _Antti Karttunen_, Jan 22 2020