OFFSET
1,1
COMMENTS
The sequence is infinite: if a number of the form p(1) * p(2) * ... * p(i)^2 * p(i+1) * ... * p(m) is in the sequence where p(1), ..., p(m) are primes, then the numbers p(1) * p(2) * ... * p(i)^q * p(i+1) * ... * p(m) are also in the sequence for q = 3, 4, ... For example, the infinite subsequence 33220, 66440, 132880, ... contains the numbers of the form 2^q * 5 * 11 * 151 for q = 2, 3, 4, ... where 2+5+11+151 = 169 = 13^2 and 2*2 + 2*5 + 2*11 + 2*151 + 5*11 + 5*151 + 11*151 = 2809 = 53^2.
In this sequence, the corresponding pairs of squares are (961, 196), (2401, 484), (900, 64), (2809, 169), (4900, 361), (7225, 729), (2304, 100), (1521, 100), (2809, 169), (7225, 729), (1225, 64), (3721, 121), (12100, 289), (4900, 361), (2704, 100), (7225, 169), (8100, 400), (2916, 169), (2809, 169), (12769, 225), (1521, 100), (7225, 729), (8464, 225), (13225, 529), (5329, 121), (3721, 121), (1369, 64), (2704, 100), (7225, 169), (13689, 289), (2809, 169), (3364, 100), (12769, 225), (12100, 289), ...
EXAMPLE
1146 = 2*3*191 is in the sequence because the divisors are {1, 2, 3, 6, 191, 382, 573, 1146}, so the sum of the semiprime divisors is 6 + 382 + 573 = 961 = 31^2 and the sum of the prime divisors is 2 + 3 + 191 = 196 = 14^2.
MAPLE
with(numtheory):for n from 2 to 310000 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if sqrt(s1)=floor(sqrt(s1)) and sqrt(s2)=floor(sqrt(s2)) then printf(`%d, `, n):else fi:od:
MATHEMATICA
Rest@ Select[Range[3*10^5], AllTrue[{DivisorSum[#, # &, PrimeOmega@ # == 2 &], DivisorSum[#, # &, PrimeQ]}, IntegerQ@ Sqrt@ # &] &] (* Michael De Vlieger, Sep 15 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jul 13 2013
STATUS
approved