A374218 -id:A374218 - 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

A374215 a(n) = A347381(n) - A374214(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 07 2024

sign

The first negative term is a(1271) = A347381(1271) - A374214(1271) = 9 - 10 = -1. The next ones occur at n=97969, 133907, 142859, 161257, 189209.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A005940, A347381, A374214, A374218 (indices of nonpositive terms).

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 || xA374215(n) = (A347381(n)-A374214(n));

A374219 Composite numbers k such that A347381 obtains the same value for all divisors of k that are larger than one, 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

15, 77, 1403, 3127, 3139, 8383, 15247, 45151, 47263, 54053, 58339, 65473, 73813, 79567, 89951, 94957, 155011, 211621, 293323, 333961, 360883, 441901, 444853, 496597, 612893, 623659, 646367, 727393, 786193, 796723, 1334083, 1456813, 1572491, 2103379, 2139793, 2477509, 2668867, 2735539, 2826787, 2903591, 3121133

Offset: 1

Antti Karttunen, Jul 07 2024

nonn

The first 47 terms are all semiprimes.
For three consecutive terms k=293323, 333961, 360883, A347381(k) = 89.
For three consecutive terms k=612893, 623659, 646367, A347381(k) = 134.

			77 has divisors [7, 11, 77] that are larger than 1. For all of them, A347381 obtains value 3, therefore 77 is included in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..47
Index entries for sequences related to sigma(n)

Cf. A000203, A005940, A347381.

Subsequence of A374218.

isA374219(n) = if(1==n || isprime(n), 0, my(w=A347381(n)); fordiv(n, d, if(d>1 && A347381(d)!=w, return(0))); (1));

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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A374215 a(n) = A347381(n) - A374214(n).

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A374219 Composite numbers k such that A347381 obtains the same value for all divisors of k that are larger than one, 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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