A327167 - OEIS

%I #7 Sep 19 2019 21:13:33

%S 1,1,2,1,1,1,1,1,3,2,1,1,1,1,8,1,1,1,1,2,2,1,1,1,6,1,5,1,1,1,1,1,2,1,

%T 2,1,1,1,2,2,1,2,1,1,12,1,1,1,3,6,2,1,1,1,2,2,2,1,1,1,1,1,12,1,1,1,1,

%U 1,2,8,1,1,1,1,48,1,2,1,1,2,7,1,1,2,2,1,2,1,1,1,2,1,2,1,1,1,1,6,3,3,1,1,1,1,128

%N a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

%H Antti Karttunen, <a href="/A327167/b327167.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A327167/a327167.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)).

%F Other identities. For all n >= 1:

%F 1+A001222(a(n)) = A327168(n).

%o (PARI)

%o A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };

%o A327167(n) = { my(m=1,v); fordiv(A276086(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

%Y Cf. A008578, A276086, A286561, A327168.

%Y Cf. also A293514, A322312, A323155, A327154, A327155, A327156.

%K nonn

%O 1,3

%A _Antti Karttunen_, Sep 19 2019