A324384 - OEIS

%I #13 Feb 26 2019 19:12:48

%S 0,1,2,1,4,1,6,1,4,1,2,11,12,1,2,1,8,1,18,1,4,7,2,1,24,1,2,1,4,1,30,1,

%T 8,1,2,7,12,1,2,1,4,1,6,1,4,1,2,1,12,1,2,1,52,1,6,1,56,1,2,1,60,1,2,1,

%U 16,1,6,1,4,1,14,1,24,1,2,1,4,1,6,1,4,1,2,1,12,1,2,1,8,1,90,1,4,1,2,1,12,1,2,1,4,1,6,1,8,1

%N a(n) = gcd(n, A276154(n)), where A276154 is the primorial base left shift.

%C For a very few primes, a(p) > 1 (then by necessity a(p) = p). In range 2 .. 2^25 there are three: 2, 11, 119039.

%H Antti Karttunen, <a href="/A324384/b324384.txt">Table of n, a(n) for n = 0..65537</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%F a(n) = gcd(n, A276154(n)).

%o (PARI)

%o A276151(n) = { my(s=1); forprime(p=2, , if(n%p, return(n-s), s *= p)); };

%o A276152(n) = { my(s=1); forprime(p=2, , if(n%p, return(s*p), s *= p)); };

%o A276154(n) = if(!n,n,(A276152(n) + A276154(A276151(n))));

%o \\ Alternatively, A276154 can be defined with A276085, A276086 and A003961:

%o A002110(n) = prod(i=1,n,prime(i));

%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961

%o A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };

%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 A276154(n) = A276085(A003961(A276086(n)));

%o A324384(n) = gcd(n,A276154(n));

%Y Cf. A002110, A049345, A276085, A276086, A276151, A276152, A276154, A323879, A324198, A324350, A324351.

%K nonn

%O 0,3

%A _Antti Karttunen_, Feb 26 2019