1,2

M. F. Hasler, Table of n, a(n) for n = 1..20000, Nov 19 2019 (first 1000 terms from Michel Marcus)

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp.

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.

R. J. Mathar, Illustrations

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.)

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.

(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

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).

Sequence in context: A058704 A316660 A098101 * A381051 A346426 A105960

Adjacent sequences: A257667 A257668 A257669 * A257671 A257672 A257673

Michel Marcus, May 03 2015

Edited by M. F. Hasler, Nov 19 2019

approved