A283157 Smallest even numbers with strictly increasing number of preimages under the sum-of-proper-divisors function.

2, 4, 6, 40, 106, 314, 1954, 2234, 2794, 11194, 22394, 58234, 111994, 160154, 291194, 425594, 560554, 1022554, 1455994, 1601594, 3203194, 11703994, 16743994, 21781754, 24751994, 53253194, 60860794, 79587194, 95295194, 181060874, 287123194, 435635194, 973772794

Offset: 1

Author

Anton Mosunov, Mar 01 2017

Keywords

• nonn

Comments

Let sigma(n) denote the sum of divisors function, and s(n):=sigma(n)-n. Let r(n) denote the number of solutions to n=s(m) and put a(1):=2. a(2) is equal to the smallest number such that r(a(2)) > r(a(1)). a(3) is equal to the smallest number such that r(a(3)) > r(a(2)), and so on.

Pomerance proved that, for every e > 0, the number of solutions to n = s(m) when n is even is O_e(n^{2/3+e}).

There are 49 elements in this sequence which do not exceed 2^40. The largest element, 690100611194, has 139 preimages.

Examples

			a(1)=2, because 2=s(m) has 0 solutions;
a(2)=4, because 4=s(9);
a(3)=6, because 6=s(6)=s(25);
a(4)=40, because 40=s(44)=s(74)=s(81);
a(5)=106, because 106=s(80)=s(104)=s(110)=s(206);
a(6)=314, because 314=s(370)=s(406)=s(442)=s(622)=s(313^2);
a(7)=1954, because 1954=s(1856)=s(1952)=s(2216)=s(2702)=s(3014)=s(3902);
a(8)=2234, because 2234=s(2536)=s(2770)=s(3454)=s(3562)=s(3706)=s(3886)=s(3922);
a(9)=2794, because 2794=s(3176)=s(3716)=s(3470)=s(3878)=s(4334)=s(4658)=s(4958)=s(4982)=s(5582).
		

Links

• Anton Mosunov, Table of n, a(n) for n = 1..49
• Kevin Chum, Richard K. Guy, Michael J. Jacobson Jr. and Anton S. Mosunov, Numerical and Statistical Analysis of Aliquot Sequences, arXiv:2110.14136 [math.NT], 2021.
• C. Pomerance, The first function and its iterates, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

Crossrefs

Cf. A005114, A283156, A287233, A287238, A287247, A287251, A287262.

Programs

PARI

v=vectorsmall(10^8);
for(n=2,#v,t=(sigma(n)-n)/2;if(denominator(t)==1 && t<=#v, v[t]++))
r=0;for(n=1,#v, if(v[n]>r,r=v[n];print1(2*n", "))) \\ Charles R Greathouse IV, Mar 02 2017

Extensions

a(20)-a(25) from Charles R Greathouse IV, Mar 02 2017

a(26)-a(31) from Anton Mosunov, Mar 03 2017

a(32)-a(49) from Anton Mosunov, Apr 20 2017