A374214 -id:A374214 - cp's OEIS frontend

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

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));

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

Original entry on oeis.org

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

A374204 a(n) is the minimum value of A347381 that it obtains among divisors of n larger than 1. By convention a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 07 2024

nonn

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

Cf. A000203, A347381, A374214.

Cf. also A374196.

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)));
A374204(n) = { my(m=-1,x); fordiv(n,d,if(d>1, x = A347381(d); if(m<0 || x

a(1) = 0, and for n > 1, a(n) = Min_{d|n, d>1} A347381(d).

For nonprime n, a(n) = min(A347381(n), A374214(n)).

A374200 a(n) is the minimum value of A347381 that it obtains among the unitary 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) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 09 2024

nonn

In contrast to A374204 and A374214 it seems that this sequence has no missing values, i.e., it is probably surjective on N.

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

Cf. A005940, A347381, A347383.

Cf. also A374204, A374214.

A374200(n) = { my(m=-1,x); fordiv(n,d,if(d>1 && 1==gcd(d,n/d), x = A347381(d); if(m<0 || x

a(1) = 0, and for n > 1, a(n) = Min_{d|n, d>1, gcd(d,n/d)=1} A347381(d).

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)}.

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

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

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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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A374204 a(n) is the minimum value of A347381 that it obtains among divisors of n larger than 1. By convention a(1) = 0.

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A374200 a(n) is the minimum value of A347381 that it obtains among the unitary 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) = 0.

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

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