A180252 Numbers where all prime divisors are of the form k^2+1.
1, 2, 4, 5, 8, 10, 16, 17, 20, 25, 32, 34, 37, 40, 50, 64, 68, 74, 80, 85, 100, 101, 125, 128, 136, 148, 160, 170, 185, 197, 200, 202, 250, 256, 257, 272, 289, 296, 320, 340, 370, 394, 400, 401, 404, 425, 500, 505, 512, 514
Offset: 1
Keywords
Examples
a(17) = 74 because 74 = 2*37 = (1^2+1)*(6^2+1).
Links
- Ivan Neretin, Table of n, a(n) for n = 1..10000
Programs
-
Maple
with(numtheory):T:=array(1..50):U:=array(1..1000):k:=1:for m from 1 to 300 do:x:=m^2+1:if type(x,prime)=true then T[k]:=x:k:=k+1:else fi:od:for x from 2 to 2000 do: B:=factorset(x):yy:=nops(B):A:=convert(T, set):if A intersect B = B then printf(`%d, `, x):else fi:od:
-
Mathematica
Select[Range@520, And @@ IntegerQ /@ Sqrt[FactorInteger[#][[All, 1]] - 1] &] (* Ivan Neretin, Aug 31 2016 *)
Formula
Sum_{n>=1} 1/a(n) = Product_{p in A002496} p/(p-1) = Product_{k in A005574} (1 + 1/k^2) = 2.809865... - Amiram Eldar, Sep 27 2020