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A374214 a(n) is the minimum value of A347381 obtained among all proper divisors of n larger than 1, where A347381 is the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree (A005940). By convention a(1) = a(p) = 0 for all primes p.

%I A374214 #17 Jul 09 2024 12:45:35
%S A374214 0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,
%T A374214 1,0,0,0,1,0,0,0,0,0,1,0,0,0,3,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,
%U A374214 1,0,0,0,0,0,1,0,3,0,0,0,1,0,0,0,1,0,1,0,0,0,2,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,3,0,3
%N A374214 a(n) is the minimum value of A347381 obtained among all proper divisors of n larger than 1, where A347381 is the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree (A005940). By convention a(1) = a(p) = 0 for all primes p.
%C A374214 It seems that values 11, 14, 16, 17, 18, 29, 40, 47 and 48 are completely missing (see the "bandgaps" in the scatter plot), most likely as they are also missing from A374481. Not so for 49, whose first occurrence is a(146507), where 146507 = 239*613. Note that A374481(112) = A374204(613) = A347381(613) = 49.
%H A374214 Antti Karttunen, <a href="/A374214/b374214.txt">Table of n, a(n) for n = 1..100000</a>
%H A374214 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A374214 For composite n, a(n) = Min_{d|n, 1<d<n} A347381(d).
%o A374214 (PARI)
%o A374214 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A374214 A252463(n) = if(!(n%2),n/2,A064989(n));
%o A374214 A347381(n) = if(1==n,0, my(lista=List([]), i, k=n, stemvec, stemlen, sbr=sigma(n)); while(k>1, listput(lista,k); k = A252463(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,sbr))>0, return(stemlen-i)); sbr = A252463(sbr)));
%o A374214 A374214(n) = { my(m=-1,x); fordiv(n,d,if(d>1 && d<n, x = A347381(d); if(m<0 || x<m, m=x))); if(-1==m,0,m); };
%Y A374214 Cf. A000004 (even bisection), A005940, A347381, A374200, A374204, A374215, A374481.
%K A374214 nonn,look
%O A374214 1,49
%A A374214 _Antti Karttunen_, Jul 07 2024