A347384 -id:A347384 - 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

A347383 Odd composites k, not powers of primes, such that for all their nontrivial unitary divisors d it holds that A347381(d) > A347381(k).

Original entry on oeis.org

189, 1271, 2125, 9261, 63767, 133907, 142859, 161257, 189209, 226967, 368063, 426373, 777923, 801727, 925101, 961193, 1003043, 4566661, 5244091, 5588327, 6031163, 6064439, 8135263, 8639879, 10074227, 10150571, 11234875, 12489107, 16016003, 19765547, 22635539

Offset: 1

Antti Karttunen, Sep 10 2021

Here nontrivial unitary divisor d of k means any divisor d|k, such that 1 < d < k and gcd(d, k/d) = 1.
Any hypothetical odd term x in A005820 (triperfect numbers) would also be a member of this sequence. Proof: such an odd number cannot be a prime power (although it must be a square), thus it must have at least two nontrivial unitary divisors (with A034444(x) >= 4). Because sigma(x) = 3*x, it must be a term of A347391. From the illustration given there, we see that any odd square y in that sequence (i.e. with A347381(y)=1) would have an abundancy index of at least three (sigma(y)/y >= 3). But because abundancy index is multiplicative and always > 1 for n > 1, any nontrivial unitary divisor d of an odd triperfect number x must have sigma(d)/d < 3, thus for all such d, A347381(d) <> 1. And neither such divisor d can be a term of A336702, because 3*x is odd, therefore we must have A347381(d) > 1 for all nontrivial unitary divisors d of such a hypothetical x.
Any odd term of A000396, i.e., an odd perfect number, if such a hypothetical number exists, must also be a term of this sequence, by reasoning similar to above. See also illustration in A347392.

			189 is a term, because A347381(189) = 1, and the only way to factor 189 into nontrivial unitary divisors is 7*27, and A347381(7) = A347381(27) = 3 > 1.
63767 = 11^2 * 17 * 31 is a term, as its nontrivial unitary divisors are [17, 31, 121, 527, 2057, 3751], at which points A347381 obtains values [6, 10, 5, 11, 6, 8], every one which is larger than A347381(63767) = 3.

Cf. A000396, A005820, A005940, A034444, A336702, A347381, A347391, A347392.

Subsequence of A347390, which is a subsequence of A347384.

isA347383(n) = if((1==n)||!(n%2)||isprimepower(n),0,my(w=A347381(n)); fordiv(n,d,if((d>1)&&(dA347381(d)<=w), return(0))); (1));

a(29)-a(31) from Jinyuan Wang, Jul 09 2025

A347390 Odd numbers k that can be factored to such a pair of coprime factors x and y that A347381(k) < min(A347381(x), A347381(y)).

Original entry on oeis.org

189, 455, 945, 1271, 1365, 2125, 4199, 6375, 9261, 12597, 13167, 15631, 18189, 20995, 21275, 24583, 26273, 29393, 30879, 42813, 43475, 46163, 46189, 46305, 46575, 46893, 54653, 63767, 63825, 65317, 67473, 67673, 73749, 78155, 78725, 89503, 90117, 90945, 92783, 93869, 106079, 108819, 119239, 122265, 127323, 129575

Offset: 1

Antti Karttunen, Sep 09 2021

nonn

			189 is a term, because A347381(189) = 1, and 189 can be factored as 7*27 with gcd(7,27)=1, and A347381(7) = A347381(27) = 3 > 1.
455 is a term, because A347381(455) = 2, and 455 can be factored as 7*65 with gcd(7,65)=1, and A347381(65) = 4 > A347381(7) = 3 > A347381(455) = 2.
945 is a term, because A347381(945) = 1, and 945 can be factored as 27*35 with gcd(27,35)=1, and A347381(27) = 3 > A347381(35) = 2 > A347381(945) = 1.
1542968918569 = (13*19*47*107)^2 is a term, because it can be factored as 893^2 * 1391^2, with gcd(893^2, 1391^2) = 1, and A347381(1391^2) = 30 > A347381(893^2) = 17 > A347381(1542968918569) = 12. (This is probably the smallest square present in the sequence).

Cf. A347381, A347391.

Subsequence of A347384. Cf. also A347383 (subsequence).

isA347390(n) = if(!(n%2),0,my(w=A347381(n)); fordiv(n,d,if(d>(n/d),return(0)); if(1==gcd(d,n/d) && (min(A347381(d),A347381(n/d))>w), return(1))); (0));

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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A347383 Odd composites k, not powers of primes, such that for all their nontrivial unitary divisors d it holds that A347381(d) > A347381(k).

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A347390 Odd numbers k that can be factored to such a pair of coprime factors x and y that A347381(k) < min(A347381(x), A347381(y)).

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