cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Antti Karttunen, Aug 30 2021

Keywords

Comments

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.

Links

Crossrefs

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.

Programs

  • PARI
    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))));
  • PARI
    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)));

Formula

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)).

Extensions

Name changed, old name is now in formula section. - Antti Karttunen, Jul 09 2024

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

Views

Author

Antti Karttunen, Jul 07 2024

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

Views

Author

Antti Karttunen, Jul 09 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A005940, A347381, A347383.
Cf. also A374204, A374214.

Programs

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

Formula

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

A374196 a(n) is the minimum value of A017666 that it obtains among divisors of n larger than 1. By convention a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 5, 1, 7, 2, 3, 2, 11, 1, 13, 2, 3, 2, 17, 1, 19, 2, 3, 2, 23, 1, 5, 2, 3, 1, 29, 1, 31, 2, 3, 2, 5, 1, 37, 2, 3, 2, 41, 1, 43, 2, 3, 2, 47, 1, 7, 2, 3, 2, 53, 1, 5, 1, 3, 2, 59, 1, 61, 2, 3, 2, 5, 1, 67, 2, 3, 2, 71, 1, 73, 2, 3, 2, 7, 1, 79, 2, 3, 2, 83, 1, 5, 2, 3, 2, 89, 1, 7, 2, 3, 2, 5, 1, 97, 2, 3, 2
Offset: 1

Views

Author

Antti Karttunen, Jul 07 2024

Keywords

Links

Crossrefs

Cf. A000203, A017666, A374198 (indices of 1's in this sequence).
Cf. also A374204.

Programs

  • PARI
    A374196(n) = { my(m=0,x); fordiv(n,d,if(d>1, x = denominator(sigma(d)/d); if(!m || x

Formula

a(1) = 1, and for n > 1, a(n) = Min_{d|n, d>1} A017666(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

Views

Author

Antti Karttunen, Jul 07 2024

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

{k | A347381(k) = A374204(k)}.
{k | A347381(k) <= A374214(k)}.

A374221 Indices of 1's in A374200, where A374200 is the minimum value of A347381 that it obtains among unitary divisors of n larger than 1.

Original entry on oeis.org

3, 4, 5, 12, 15, 20, 21, 24, 33, 35, 36, 39, 40, 44, 45, 48, 51, 52, 55, 57, 60, 65, 68, 69, 75, 76, 80, 85, 87, 92, 93, 95, 96, 100, 105, 108, 111, 115, 116, 120, 123, 124, 129, 132, 135, 141, 145, 147, 148, 155, 156, 159, 160, 164, 165, 168, 172, 177, 180, 183, 185, 188, 189, 192, 195, 196, 201, 204, 205, 212, 213
Offset: 1

Views

Author

Antti Karttunen, Jul 09 2024

Keywords

Comments

It seems that for all odd terms x in this sequence, A374204(x) = 1, and they form a subsequence of all terms k for which A374204(k) = 1, that might be sequence A086748.

Links

Crossrefs

Cf. A000203, A005940, A347381, A374200, A374204, A374220 (characteristic function).
Cf. also A086748 and A347391 (very likely a subsequence).

Programs

Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.