A360111 - OEIS

%I #42 Feb 07 2024 20:39:10

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

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

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

%N a(n) = 1 if there is no prime p such that p^p divides n, but for the arithmetic derivative of n such a prime exists, otherwise a(n) = 0; a(1) = 0 by convention.

%C Question: What is the asymptotic mean of this and sequences like A341996 and its complement A368915, and related A368916? - _Antti Karttunen_, Jan 11 2024

%C Question: Is the asymptotic mean of this sequence 1 - Product_{p prime} (1 - 1/p^(1+p)) = 0.13585792767780221591...? (I.e. complementary to that of A368916). But see also A368911. - _Antti Karttunen_, Jan 29 2024

%H Antti Karttunen, <a href="/A360111/b360111.txt">Table of n, a(n) for n = 1..100000</a>

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

%F a(n) = A359550(n) * A341996(n).

%F a(n) = [-1 == A256750(n)], where [ ] is the Iverson bracket.

%F For n > 1, a(n) = A359550(n) - A368915(n). - _Antti Karttunen_, Jan 11 2024

%e a(12) = 0, because for both 12 and 12' = A003415(12) = 16 there is a prime p (in both cases p=2) such that p^p divides them.

%e a(15) = 1, because 15 = 3*5 has no such prime divisor p that p^p would divide it, while 15' = 8 is divisible by 2^2.

%t d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); q[n_] := AllTrue[FactorInteger[n], Last[#] < First[#] &]; q[1] = True; a[1] = 0; a[n_] := If[q[n] && ! q[d[n]], 1, 0]; Array[a, 100] (* _Amiram Eldar_, Jan 31 2023 *)

%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 A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

%o A360111(n) = ((n>1)&&A359550(n)&&!A359550(A003415(n)));

%Y Cf. A003415, A256750, A341996, A359550, A368911, A368915, A368916.

%Y Characteristic function of A327934.

%Y After n=1 differs from A360109 for the next time at n=81, where a(81) = 0, while A360109(81) = 1.

%Y Differs from A353479 for the first time at n=158, where a(158) = 1, while A353479(158) = 1.

%K nonn

%O 1

%A _Antti Karttunen_, Jan 31 2023