A374481 -id:A374481 - cp's OEIS frontend

A347381 Distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree (A005940).

0, 0, 1, 1, 1, 0, 3, 2, 2, 3, 3, 2, 2, 3, 1, 3, 6, 3, 5, 1, 4, 5, 7, 2, 3, 4, 3, 0, 8, 4, 10, 4, 4, 7, 2, 4, 4, 7, 3, 4, 10, 4, 9, 4, 3, 9, 13, 4, 4, 4, 7, 7, 15, 4, 5, 5, 6, 9, 15, 4, 7, 10, 3, 5, 4, 6, 12, 6, 8, 5, 19, 5, 9, 6, 4, 8, 3, 5, 19, 4, 3, 11, 20, 4, 7, 11, 9, 6, 22, 4, 4, 8, 11, 15, 7, 5, 24, 5, 3, 5, 20

Offset: 1

Antti Karttunen, Aug 30 2021

a(n) tells about the degree of relatedness between n and sigma(n) in Doudna tree (see the illustration in A005940). It is 0 for those n where sigma(n) is one of the descendants of n, 1 for those n where the nearest common ancestor of n and sigma(n) is the parent of n, 2 for those n where the nearest common ancestor of n and sigma(n) is the grandparent of n, and so on.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Indices of 0 .. 5 in this sequence are given by {2} U A336702, A347391, A347392, A347393, A347394, A374465.
Cf. A000203, A027687, A156552, A252463, A252464, A332221, A347380, A347383, A347384, A347390, A374481 [a(prime(n))], A374482 (indices of records), A374483 (record values).
Cf. A374200, A374204, A374214, A374218, A374219.
Cf. also A336834.

A000523(n) = logint(n,2);
Abincompreflen(x, y) = if(!x || !y, 0, my(xl=A000523(x), yl=A000523(y), s=min(xl,yl), k=0); x >>= (xl-s); y >>= (yl-s); while(s>=0 && !bitand(1,bitxor(x>>s,y>>s)), s--; k++); (k));
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
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = if(1==n,0,(bigomega(n) + A061395(n) - 1));
A347381(n) = (A252464(n)-Abincompreflen(A156552(n), A156552(sigma(n))));

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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
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)));

a(n) = A252464(n) - A347380(n), where A347380(n) is the length of the common prefix in binary expansions of A156552(n) and A332221(n) = A156552(sigma(n)).

Name changed, old name is now in formula section. - Antti Karttunen, Jul 09 2024

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.

Original entry on oeis.org

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, 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, 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

Offset: 1

Antti Karttunen, Jul 07 2024

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.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to sigma(n)

Cf. A000004 (even bisection), A005940, A347381, A374200, A374204, A374215, A374481.

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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
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)));
A374214(n) = { my(m=-1,x); fordiv(n,d,if(d>1 && dA347381(d); if(m<0 || x

For composite n, a(n) = Min_{d|n, 1A347381(d).

A374482 Indices of records in A347381, where A347381 is the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree (A005940).

Original entry on oeis.org

1, 3, 7, 17, 23, 29, 31, 47, 53, 71, 83, 89, 97, 107, 127, 139, 149, 167, 179, 191, 199, 233, 239, 241, 269, 293, 337, 349, 359, 373, 383, 419, 431, 449, 461, 479, 499, 569, 571, 577, 587, 599, 643, 647, 701, 719, 769, 809, 839, 857, 863, 881, 967, 971, 1013, 1039, 1049, 1091, 1151, 1187, 1217, 1223, 1231, 1249, 1279

Offset: 1

Antti Karttunen, Jul 09 2024

nonn

Most likely, all terms are primes after the initial 1. Mersenne primes (A000668) are a subsequence.

Antti Karttunen, Table of n, a(n) for n = 1..18144
Index entries for sequences related to sigma(n)

Cf. A000668 (subsequence), A005940, A347381, A374483 (record values).

Cf. also A374481.

cp's OEIS Frontend

A347381 Distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree (A005940).

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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.

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A374482 Indices of records in A347381, where A347381 is the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree (A005940).

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