OFFSET
1,1
COMMENTS
Any term x = a(m) can be combined with any term y = A275701(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2. Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one. So far, these two sequences have produced only one amicable pair: (1210,1184) = (a(5),A275701(2)) = (A063990(4),A063990(3)). If more are ever found, then they will also exhibit x-y = 26.
Notice that:
a(1) = 58 = 29* 2 = (4^1+25)*(4^1)/2
a(3) = 328 = 41* 8 = (4^2+25)*(4^2)/2
a(6) = 2848 = 89* 32 = (4^3+25)*(4^3)/2
a(7) = 35968 = 281*128 = (4^4+25)*(4^4)/2
a(8) = 537088 = 1049*512 = (4^5+25)*(4^5)/2.
If p = 4^k+25 is prime and n = p*(p-25)/2, then it is not hard to show that 2*n - sigma(n) = 26. The values of k in A204388 will guarantee that p is prime (A104072). Similarly, if q = 2*4^k+25 is prime and n = q*(q-25)/2, then 2*n - sigma(n) = 26. However, q will never be prime since it will always be divisible by 3: 2*4^k+25 == (2*1^k+25) mod 3 == 27 mod 3 == 0 mod 3. So, the following values will be in this sequence and provide upper bounds for the next seven terms:
(4^10+25)*(4^10)/2 = 549768921088 >= a(9)
(4^11+25)*(4^11)/2 = 8796145451008 >= a(10)
(4^17+25)*(4^17)/2 = 147573952804424777728 >= a(11)
(4^35+25)*(4^35)/2 = 696898287454081973187748591279228938354688 >= a(12)
(4^46+25)*(4^46)/2 = 12259964326927110866866776279099475433218926722425028608 >= a(13)
(4^56+25)*(4^56)/2 = 13479973333575319897333507543509880240529303896615642871755920375808 >= a(14)
(4^59+25)*(4^59)/2 = 55213970774324510299478046898216207773446358605225195265697257166471168 >= a(15).
The rightmost digit of n = p*(p-25)/2 will always be 8. [Proof: If k is odd, then 4^k+25 == 9 mod 10 and (4^k)/2 == 2 mod 10, which implies that p*(p-25)/2 == 8 mod 10. If k is even, then 4^k+25 == 1 mod 10 and (4^k)/2 == 8 mod 10, which implies that p*(p-25)/2 == 8 mod 10.]
a(10) > 2.3*10^12. - Giovanni Resta, Aug 07 2016
a(11) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
LINKS
EXAMPLE
a(1) = 58, since 2*58-sigma(58) = 116-90 = 26.
a(2) = 75, since 2*75-sigma(75) = 150-124 = 26.
a(3) = 328, since 2*328-sigma(328) = 656-630 = 26.
MATHEMATICA
Select[Range[10^6], 2 # - (DivisorSigma[1, #]) == 26 &] (* Vincenzo Librandi, Aug 06 2016 *)
PROG
(Magma) [n: n in [1..2*10^6] | (2*n-SumOfDivisors(n)) eq 26]; // Vincenzo Librandi, Aug 06 2016
(PARI) is(n) = 2*n-sigma(n)==26 \\ Felix Fröhlich, Aug 06 2016
CROSSREFS
KEYWORD
nonn,more,changed
AUTHOR
Timothy L. Tiffin, Aug 05 2016
EXTENSIONS
a(9) from Giovanni Resta, Aug 07 2016
a(10) from Hiroaki Yamanouchi, Aug 21 2018
STATUS
approved