1,1

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.

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.

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).

(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

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

Sequence in context: A269265 A173818 A263541 * A344422 A084324 A329529

Adjacent sequences: A283154 A283155 A283156 * A283158 A283159 A283160

Anton Mosunov, Mar 01 2017

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

approved