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A356157 The nearest common ancestor of sigma(n) and gcd(n, sigma(n)) in the Doudna tree (A005940).

%I A356157 #11 Aug 04 2022 22:27:19
%S A356157 1,1,1,1,1,6,1,1,1,2,1,2,1,2,3,1,1,3,1,2,1,2,1,6,1,2,1,28,1,2,1,1,3,2,
%T A356157 1,1,1,2,1,3,1,6,1,2,3,2,1,2,1,1,2,2,1,6,1,2,1,2,1,3,1,2,1,1,1,2,1,2,
%U A356157 3,2,1,3,1,2,1,2,1,3,1,2,1,2,1,28,1,2,3,2,1,2,7,2,1,2,3,3,1,1,3,1,1,2,1,2,3
%N A356157 The nearest common ancestor of sigma(n) and gcd(n, sigma(n)) in the Doudna tree (A005940).
%H A356157 Antti Karttunen, <a href="/A356157/b356157.txt">Table of n, a(n) for n = 1..65537</a>
%H A356157 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A356157 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%o A356157 (PARI)
%o A356157 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 A356157 Abinprefix(n,k) = { my(digs=binary(n)); fromdigits(vector(k,i,digs[i]),2); };
%o A356157 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
%o A356157 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 A356157 A348040sq(x,y) = Abincompreflen(A156552(x), A156552(y));
%o A356157 A348041sq(x,y) = A005940(1+Abinprefix(A156552(x),A348040sq(x,y)));
%o A356157 A356157(n) = A348041sq(sigma(n),gcd(n, sigma(n)));
%Y A356157 Cf. A000203, A009194, A336702 (fixed points), A348040, A348041.
%Y A356157 Cf. also A347879, A356156, A356307.
%K A356157 nonn
%O A356157 1,6
%A A356157 _Antti Karttunen_, Jul 30 2022