|
|
A283157
|
|
Smallest even numbers with strictly increasing number of preimages under the sum-of-proper-divisors function.
|
|
7
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
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).
|
|
PROG
|
(PARI) v=vectorsmall(10^8);
for(n=2, #v, t=(sigma(n)-n)/2; if(denominator(t)==1 && t<=#v, v[t]++))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|