A347381 -id:A347381 - cp's OEIS frontend

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

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

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)).

A374214 a(n) is the minimum value of A347381 obtained among all proper 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) = a(p) = 0 for all primes p.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 3, 0, 3

Offset: 1

Antti Karttunen, Jul 07 2024

It seems that values 11, 14, 16, 17, 18, 29, 40, 47 and 48 are completely missing (see the "bandgaps" in the scatter plot), most likely as they are also missing from A374481. Not so for 49, whose first occurrence is a(146507), where 146507 = 239*613. Note that A374481(112) = A374204(613) = A347381(613) = 49.

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

Cf. A000004 (even bisection), A005940, A347381, A374200, A374204, A374215, A374481.

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 || x

For composite n, a(n) = Min_{d|n, 1A347381(d).

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).

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.

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));

A347393 Positions of 3's in A347381.

Original entry on oeis.org

7, 10, 11, 14, 16, 18, 25, 27, 39, 45, 63, 77, 81, 99, 105, 135, 182, 270, 819, 1365, 1392, 1638, 4250, 15631, 21275, 63767, 122944, 161257, 203203, 446369, 936100, 1128799, 1773827, 2808300

Offset: 1

Antti Karttunen, Aug 31 2021

nonn

Cf. A347381.

Cf. also A046060, A336702, A347391, A347392, A347394.

A347394 Positions of 4's in A347381.

Original entry on oeis.org

21, 26, 30, 32, 33, 36, 37, 40, 42, 44, 48, 49, 50, 54, 60, 65, 75, 80, 84, 90, 91, 112, 120, 125, 126, 153, 162, 175, 176, 198, 208, 220, 231, 252, 272, 275, 304, 325, 343, 368, 400, 425, 475, 546, 575, 725, 765, 775, 11132, 12750, 13167, 31262, 46893, 55660, 63825, 78155, 93500, 171171, 191301, 406406, 483771, 609609

Offset: 1

Antti Karttunen, Aug 31 2021

nonn

Question: Why the sudden rarification of terms after a(48) = 775?

Cf. A347381.

Cf. also A336702, A347391, A347392, A347393.

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)}.

A374483 Record values 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

0, 1, 3, 6, 7, 8, 10, 13, 15, 19, 20, 22, 24, 27, 30, 32, 34, 36, 39, 41, 45, 46, 50, 52, 55, 61, 67, 68, 70, 71, 74, 79, 82, 86, 87, 90, 94, 98, 103, 105, 106, 108, 111, 117, 121, 126, 134, 138, 144, 146, 149, 151, 162, 163, 169, 171, 174, 179, 189, 191, 192, 194, 200, 203, 204, 208, 212, 220, 221, 224, 226, 229, 231

Offset: 1

Antti Karttunen, Jul 09 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..18144
Index entries for sequences related to sigma(n)

Cf. A005940, A347381, A374482.

a(n) = A347381(A374482(n)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A374214 a(n) is the minimum value of A347381 obtained among all proper 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) = a(p) = 0 for all primes p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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

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)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A347393 Positions of 3's in A347381.

Views

Author

Keywords

Links

Crossrefs

A347394 Positions of 4's in A347381.

Views

Author

Keywords

Comments

Links

Crossrefs

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI