A369961 - OEIS

%I #9 Feb 07 2024 20:39:38

%S 0,1,1,0,0,1,1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,

%T 1,0,0,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,0,0,1,1,0,1,1,0,0,1,1,1,

%U 0,0,0,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0,0,0,1,1,0,1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0,0,1,0,0,0,0,1,0

%N a(n) = 1 if gcd(n, A003415(n)) is equal to gcd(n, A276086(n)), otherwise 0, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

%H Antti Karttunen, <a href="/A369961/b369961.txt">Table of n, a(n) for n = 0..100000</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

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

%F a(n) = [A085731(n) == A324198(n)], where [ ] is the Iverson bracket.

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o A369961(n) = (gcd(n,A003415(n))==gcd(n,A276086(n)));

%o (PARI)

%o A085731(n) = { my(f=factor(n)); for(i=1, #f~, if (f[i, 2] % f[i, 1], f[i, 2]--); ); factorback(f); };

%o A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };

%o A369961(n) = ((n>0) && (A085731(n)==A324198(n)));

%Y Characteristic function of A369962.

%Y Cf. A003415, A083345, A085731, A276086, A324198, A351251.

%K nonn

%O 0

%A _Antti Karttunen_, Feb 07 2024