A374219 -id:A374219 - 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

A374218 After the initial 1, numbers k such that A347381 obtains its minimum value at k, of all the divisors d of k 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

1, 2, 6, 15, 28, 77, 189, 496, 899, 945, 1271, 1403, 2125, 3127, 3139, 6375, 8128, 8383, 9261, 13843, 15247, 15631, 30240, 32760, 45151, 46305, 47263, 54053, 54653, 58339, 63767, 65473, 73813, 79567, 89951, 92783, 94957, 97969, 133907, 142859, 155011, 161257, 189209, 211621, 249001, 293323, 333961, 360883, 368063

Offset: 1

Antti Karttunen, Jul 07 2024

nonn

Not all terms of A347383 are included here. The first missing ones are 226967, 925101, 961193, 4566661, 6031163, 6064439, 11234875.

			189 has divisors 3, 7, 9, 21, 27, 63, 189 larger than 1. A347381 applied to them gives 1, 3, 2, 4, 3, 3, 1, so the largest divisor 189 gets minimal value 1 (which also occurs at the smallest prime divisor 3), thus 189 is included in this sequence.
496 has divisors 2, 4, 8, 16, 31, 62, 124, 248, 496 larger than 1. A347381 applied to them gives 0, 1, 2, 3, 10, 10, 9, 12, 0, so the largest divisor 496 gets minimal value 0 (which also occurs at the smallest prime divisor 2), thus 496 is included in this sequence.
1271 has divisors 31, 41, 1271 larger than 1. A347381 applied to them gives 10, 10, 9, of which minimal value 9 occurs at the largest divisor (1271 itself), thus 1271 is included in this sequence.

Cf. A347381, A347383, A374204, A374214.
Indices of nonpositive terms in A374215.
Cf. A336702, A374219 (subsequences).

isA374218(n) = if(n>2 && isprime(n), 0, my(w=A347381(n)); fordiv(n, d, if(d>1 && dA347381(d)

{k | A347381(k) = A374204(k)}.

{k | A347381(k) <= A374214(k)}.

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

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A374218 After the initial 1, numbers k such that A347381 obtains its minimum value at k, of all the divisors d of k 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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