A331595 - OEIS

%I #11 Jan 29 2020 02:57:47

%S 1,2,4,3,8,3,16,5,3,3,32,5,64,3,18,7,128,15,256,5,18,3,512,7,3,3,5,5,

%T 1024,15,2048,11,18,3,18,7,4096,3,18,7,8192,15,16384,5,50,3,32768,11,

%U 3,45,18,5,65536,7,108,7,18,3,131072,7,262144,3,50,13,108,15,524288,5,18,45,1048576,11,2097152,3,15,5,18,15,4194304,11,7,3

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

%H Antti Karttunen, <a href="/A331595/b331595.txt">Table of n, a(n) for n = 1..16384</a>

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

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

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

%t Array[If[# == 1, 1, GCD @@ {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[#]] &, 82] (* _Michael De Vlieger_, Jan 24 2020, after _JungHwan Min_ at A122111 *)

%o (PARI)

%o A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};

%o A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));

%o A241909(n) = if(1==n||isprime(n),2^primepi(n),my(f=factor(n),h=1,i,m=1,p=1,k=1); while(k<=#f~, p = nextprime(1+p); i = primepi(f[k,1]); m *= p^(i-h); h = i; if(f[k,2]>1, f[k,2]--, k++)); (p*m));

%o A331595(n) = gcd(A122111(n), A241909(n));

%Y Cf. A122111, A241909, A241916, A331596 (number of distinct prime factors), A331597, A331598, A331599, A331600.

%Y Cf. also A280489, A280491.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jan 22 2020