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A356156 The nearest common ancestor of n and gcd(n, sigma(n)) in the Doudna tree (A005940).

%I A356156 #13 Aug 04 2022 22:27:27
%S A356156 1,1,1,1,1,6,1,1,1,2,1,2,1,2,3,1,1,2,1,2,1,2,1,12,1,2,1,28,1,6,1,1,3,
%T A356156 2,1,1,1,2,1,10,1,3,1,2,3,2,1,2,1,1,3,2,1,2,1,2,1,2,1,6,1,2,1,1,1,3,1,
%U A356156 2,3,2,1,2,1,2,1,2,1,3,1,2,1,2,1,5,1,2,3,2,1,2,5,2,1,2,5,12,1,1,3,1,1,3,1,2,3
%N A356156 The nearest common ancestor of n and gcd(n, sigma(n)) in the Doudna tree (A005940).
%H A356156 Antti Karttunen, <a href="/A356156/b356156.txt">Table of n, a(n) for n = 1..65537</a>
%H A356156 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A356156 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A356156 a(n) = A348041(n, A009194(n)) = A348041(n, gcd(n, A000203(n))).
%o A356156 (PARI)
%o A356156 Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
%o A356156 Abinprefix(n,k) = { my(digs=binary(n)); fromdigits(vector(k,i,digs[i]),2); };
%o A356156 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
%o A356156 A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
%o A356156 A348040sq(x,y) = Abincompreflen(A156552(x), A156552(y));
%o A356156 A348041sq(x,y) = A005940(1+Abinprefix(A156552(x),A348040sq(x,y)));
%o A356156 A356156(n) = A348041sq(n,gcd(n, sigma(n)));
%Y A356156 Cf. A000203, A007691 (fixed points), A009194, A348040, A348041.
%Y A356156 Cf. also A347879, A356157, A356158, A356306.
%K A356156 nonn
%O A356156 1,6
%A A356156 _Antti Karttunen_, Jul 30 2022