cp's OEIS Frontend

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

%I A283157 #29 Oct 28 2021 02:00:41
%S A283157 2,4,6,40,106,314,1954,2234,2794,11194,22394,58234,111994,160154,
%T A283157 291194,425594,560554,1022554,1455994,1601594,3203194,11703994,
%U A283157 16743994,21781754,24751994,53253194,60860794,79587194,95295194,181060874,287123194,435635194,973772794
%N A283157 Smallest even numbers with strictly increasing number of preimages under the sum-of-proper-divisors function.
%C A283157 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.
%C A283157 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}).
%C A283157 There are 49 elements in this sequence which do not exceed 2^40. The largest element, 690100611194, has 139 preimages.
%H A283157 Anton Mosunov, <a href="/A283157/b283157.txt">Table of n, a(n) for n = 1..49</a>
%H A283157 Kevin Chum, Richard K. Guy, Michael J. Jacobson Jr. and Anton S. Mosunov, <a href="https://arxiv.org/abs/2110.14136">Numerical and Statistical Analysis of Aliquot Sequences</a>, arXiv:2110.14136 [math.NT], 2021.
%H A283157 C. Pomerance, <a href="https://math.dartmouth.edu/~carlp/aliquot.pdf">The first function and its iterates</a>, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.
%e A283157 a(1)=2, because 2=s(m) has 0 solutions;
%e A283157 a(2)=4, because 4=s(9);
%e A283157 a(3)=6, because 6=s(6)=s(25);
%e A283157 a(4)=40, because 40=s(44)=s(74)=s(81);
%e A283157 a(5)=106, because 106=s(80)=s(104)=s(110)=s(206);
%e A283157 a(6)=314, because 314=s(370)=s(406)=s(442)=s(622)=s(313^2);
%e A283157 a(7)=1954, because 1954=s(1856)=s(1952)=s(2216)=s(2702)=s(3014)=s(3902);
%e A283157 a(8)=2234, because 2234=s(2536)=s(2770)=s(3454)=s(3562)=s(3706)=s(3886)=s(3922);
%e A283157 a(9)=2794, because 2794=s(3176)=s(3716)=s(3470)=s(3878)=s(4334)=s(4658)=s(4958)=s(4982)=s(5582).
%o A283157 (PARI) v=vectorsmall(10^8);
%o A283157 for(n=2,#v,t=(sigma(n)-n)/2;if(denominator(t)==1 && t<=#v, v[t]++))
%o A283157 r=0;for(n=1,#v, if(v[n]>r,r=v[n];print1(2*n", "))) \\ _Charles R Greathouse IV_, Mar 02 2017
%Y A283157 Cf. A005114, A283156, A287233, A287238, A287247, A287251, A287262.
%K A283157 nonn
%O A283157 1,1
%A A283157 _Anton Mosunov_, Mar 01 2017
%E A283157 a(20)-a(25) from _Charles R Greathouse IV_, Mar 02 2017
%E A283157 a(26)-a(31) from _Anton Mosunov_, Mar 03 2017
%E A283157 a(32)-a(49) from _Anton Mosunov_, Apr 20 2017