A374483 -id:A374483 - 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

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.

A374481 The distance from prime(n) to the nearest common ancestor of prime(n) and 1+prime(n) in the Doudna-tree (A005940).

Original entry on oeis.org

0, 1, 1, 3, 3, 2, 6, 5, 7, 8, 10, 4, 10, 9, 13, 15, 15, 7, 12, 19, 9, 19, 20, 22, 24, 20, 21, 27, 26, 23, 30, 28, 25, 32, 34, 28, 15, 25, 36, 31, 39, 39, 41, 19, 41, 45, 31, 44, 42, 43, 46, 50, 52, 51, 42, 52, 55, 51, 25, 46, 41, 61, 61, 59, 28, 51, 44, 67, 60, 68, 55, 70, 64, 71, 69, 74, 73, 32, 61, 69, 79, 35, 82

Offset: 1

Antti Karttunen, Jul 09 2024

nonn

Question: Is there any reasonable lower bound for this sequence?

Considering k that do not occur as terms of this sequence, see also A374214.

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

Cf. A000040, A000203, A005940, A059305, A241917, A252464, A319988, A348040, A347381.

Cf. also A374214, A374482, A374483.

PARI
```
A374481(n) = A347381(prime(n));
```

A241917(n) = if(isprime(n), primepi(n), if(1>=omega(n), 0, my(f=factor(n)); if(f[#f~,2]>1, 0, primepi(f[#f~,1])-primepi(f[(#f~)-1,1]))));
A374481(n) = if(1==n,0,(-1+n-A241917(1+prime(n))));

a(n) = A347381(A000040(n)) = n - A348040(A000040(n), 1+A000040(n)).
For all n >= 1, a(A059305(n)) = A059305(n)-1.
If A319988(1+A000040(n)) then a(n) = n-1.
For n > 1, a(n) = n - A241917(1+prime(n)) - 1. - Peter Munn and Antti Karttunen, Jul 10 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

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).

Views

Author

Keywords

Comments

Links

Crossrefs

A374481 The distance from prime(n) to the nearest common ancestor of prime(n) and 1+prime(n) in the Doudna-tree (A005940).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula