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.

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A257348 Repeatedly applying the map x -> sigma(x) partitions the natural numbers into a number of disjoint trees; sequence gives the (conjectural) list of minimal representatives of these trees.

Original entry on oeis.org

1, 2, 5, 16, 19, 27, 29, 33, 49, 50, 52, 66, 81, 85, 105, 146, 147, 163, 170, 189, 197, 199, 218, 226, 243, 262, 303, 315, 343, 424, 430, 438, 453, 461, 463, 469, 472, 484, 489, 513, 530, 550, 584, 677, 722, 746, 786, 787, 804, 813, 821, 831, 842, 859, 867, 876, 892, 903, 914, 916, 937, 977, 982, 988, 990, 1029
Offset: 1

Views

Author

N. J. A. Sloane, May 01 2015, following a suggestion from Kerry Mitchell

Keywords

Comments

Very little is known for certain. Even the trajectories of 2 (A007497) and 5 (A051572) under repeated application of the map x -> sigma(x) (cf. A000203) are only conjectured to be disjoint.
The thousand-term b-file (up to 141441) has been checked to correspond to disjoint trees for 265 iterations of sigma on each term, and every non-term n < 141441 merges (in at most 21 iterations) with an earlier iteration sequence. - Hans Havermann, Nov 22 2019
Rather than trees we mean connected components of the graphs with edges x -> sigma(x). The number 1 is a fixed point, i.e., a cycle of length 1 under iterations of sigma, it is not part of a tree. But since sigma(n) > n for n > 1 there are no other cycles. - M. F. Hasler, Nov 21 2019

References

  • Kerry Mitchell, Posting to Math Fun Mailing List, Apr 30 2015

Links

Crossrefs

Cf. A000203 (sigma), A007497 (trajectory of 2), A051572 (trajectory of 5), A257349 (trajectory of 16).
Cf. A216200 (number of disjoint trees up to n); A257669 and A257670: size and smallest number of subtree rooted in n.

Extensions

More terms from Hans Havermann, May 02 2015

A257670 Minimum term in the sigma(x) -> x subtree whose root is n.

Original entry on oeis.org

1, 2, 2, 2, 5, 5, 2, 2, 9, 10, 11, 5, 9, 9, 2, 16, 17, 10, 19, 19, 21, 22, 23, 2, 25, 26, 27, 5, 29, 29, 16, 16, 33, 34, 35, 22, 37, 37, 10, 27, 41, 19, 43, 43, 45, 46, 47, 33, 49, 50, 51, 52, 53, 34, 55, 5, 49, 58, 59, 2, 61, 61, 16, 64, 65, 66, 67, 67, 69
Offset: 1

Views

Author

Michel Marcus, May 03 2015

Keywords

Examples

			We have the following trees (a <- b means sigma(a) = b):
  2 <-- 3 <-- 4 <-- 7 <-- 8 <-- 15 <-- 24 <-- 60 <-- ...
                    9 <-- 13 <-- 14 <-’
  5 <-- 6 <-- 12 <-- 28 <-- 56 <-- 120 <-- ...
        11 <-’             /
       10 <-- 18 <-- 39 <-’
The number 1 has strictly speaking an arrow to itself, so it is not part of a tree. (For all n > 1, sigma(n) > n, so no other fixed point or longer "cycle" can exist.) But actually we rather consider connected components, and let a(1) = 1 as the smallest element of this connected component.
a(2) = 2, since there is no smaller x such that sigma(x) = 2: the subtree with root 2 is reduced to a single node: 2. Similarly, a(m) = m for all m in A007369.
For n=3, since sigma(2) = 3, the tree whose root is 3 has 2 nodes: 2 and 3, and the smallest one is 2, hence a(3) = 2.
Similarly, although 24 occurs directly first at sigma(14), it is also reached from 15 which is in turn reached, via intermediate steps, from 2. Thus, the subtree with root 24 has as 2 as smallest element, whence a(24) = 2.

Links

Crossrefs

Cf. A000203 (sigma), A007369 (sigma(x) = n has no solution).
Cf. A216200 (number of disjoint trees), A257348 (minimal node of all trees).
Cf. A257669 (number of terms in current tree).

Programs

  • PARI
    lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, my(s = i); while (s <= nn, if (v[s] == 0, v[s] = i); s = sigma(s););); for (i=1, nn, if (v[i] == 0, v[i] = i);); v;} \\ Michel Marcus, Nov 19 2019
  • PARI
    A257670(n)=if(n>2,vecmin(concat(apply(self,invsigma(n)),n)),n) \\ See Alekseyev-link for invsigma(). - David A. Corneth and M. F. Hasler, Nov 20 2019

Formula

a(m) = m for m in A007369: sigma(x) = m has no solution. [Corrected by M. F. Hasler, Nov 19 2019]
a(A007497(n)) = 2; a(A051572(n)) = 5; a(A257349(n)) = 16. (These sequences being the trajectory of 2, 5 resp. 16 under iterations of sigma = A000203.)

Extensions

Edited by M. F. Hasler, Nov 19 2019

A309727 a(n) is the least integer k such that for some iteration of sigma applied to k, one gets the n-th term of A002191, the list of possible values for the function sum of divisors.

Original entry on oeis.org

1, 2, 2, 5, 2, 2, 5, 9, 9, 2, 10, 19, 2, 5, 29, 16, 16, 22, 37, 10, 27, 19, 43, 33, 34, 5, 49, 2, 61, 16, 67, 29, 73, 45, 49, 43, 27, 22, 50, 19, 52, 101, 16, 85, 109, 22, 73, 5, 81, 33, 67, 64, 50, 86, 81, 137, 76, 66, 149, 111, 99, 157, 81, 106, 163, 2, 52, 173, 129
Offset: 1

Views

Author

Michel Marcus, Oct 14 2019

Keywords

Comments

The set union of this sequence is 1 U A007369.

Examples

			For n = 5, A002191(5) is 7, and 4 iterations of sigma applied to 2 give 7, and no integer less than 2 will give 7, so a(5)=2.

Links

Crossrefs

Cf. A000203, A002191, A007369, A007497, A051572, A257349.
A257670 is a better version for this sequence.

Programs

  • PARI
    list(lim) = select(n->n<=lim, Set(vector(lim\=1, n, sigma(n))));
lista(nn) = {my(vs = list(nn), v = vector(#vs)); v[1] = 1; for (n=2, #vs, for (k=2, vs[n], my(kk=k); while (sigma(kk) <= vs[n], kk=sigma(kk)); if (kk == vs[n], v[n] = k; break););); v;}

Formula

a(n) = 2 when A002191(n) is in A007497.
a(n) = 5 when A002191(n) is in A051572.
a(n) = 16 when A002191(n) is in A257349.
Showing 1-3 of 3 results.

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