A257669 Number of terms in the sigma(x) -> x subtree whose root is n.

1, 1, 2, 3, 1, 2, 4, 5, 1, 1, 1, 4, 2, 3, 6, 1, 1, 3, 1, 2, 1, 1, 1, 11, 1, 1, 1, 5, 1, 2, 3, 5, 1, 1, 1, 2, 1, 2, 4, 2, 1, 5, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 10, 2, 1, 1, 15, 1, 2, 6, 1, 1, 1, 1, 2, 1, 1, 1, 7, 1, 2, 1, 1, 1, 2, 1, 4, 1, 1, 1, 5, 1, 1

Offset: 1

Author

Michel Marcus, May 03 2015

Keywords

• nonn

Comments

For terms m of A007369, numbers m such that sigma(x) = m has no solution, as well as for m = 1, a(m) = 1.

See A257670 for more information, examples, etc. - M. F. Hasler, Nov 19 2019

Records are: a(1) = 1 = a(2), a(3) = 2, a(4) = 3, a(7) = 4, a(8) = 5, a(15) = 6, a(24) = 11, a(60) = 15, a(120) = 16, a(168) = 22 = a(336), a(360) = 26, a(480) = 39, a(1344) = 43, a(1512) = 54, a(1920) = 57, a(2016) = 65, a(2880) = 70, a(4800) = 80, a(5040) = 88, a(6552) = 93, a(8064) = 125, ... - M. F. Hasler, Nov 20 2019

Examples

			For n = 2, a(2) = 1, since there is no x such that sigma(x) = 2, so the subtree with root 2 is reduced to a single node: 2.
For n = 3, since sigma(2) = 3, the tree with root 3 has 2 nodes: 2 and 3, hence a(3) = 2.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000, Nov 20 2019
• M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invsigma.gp, Oct. 2005
• G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.

Crossrefs

Cf. A007369 (sigma(x) = n has no solution).

Cf. A216200 (number of disjoint trees), A257348 (minimal nodes of all trees).

Cf. A257670 (minimal representative of current tree).

Programs

PARI

A257669_vec(N)={my(C=Map(),s,c); vector(N,n,mapput(C,s=sigma(n), if(mapisdefined(C,s), mapget(C,s))+ c=if(mapisdefined(C,n), mapget(C,n) + mapdelete(C,n))+1);c)} \\ M. F. Hasler, Nov 20 2019

PARI

apply( A257669(n)=if(n>1,vecsum(apply(self,invsigma(n))))+1, [1..99]) \\ See Alekseyev-link for invsigma(). - M. F. Hasler, Nov 20 2019, replacing earlier code from Michel Marcus

Formula

a(A007369(n)) = 1.