A275702 Numbers n whose deficiency is 26: 2n - sigma(n) = 26.

58, 75, 328, 850, 1210, 2848, 35968, 537088, 549768921088, 8796145451008

Offset: 1

Author

Timothy L. Tiffin, Aug 05 2016

Keywords

• nonn
• more

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

Examples

			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.
		

Links

• D. Alpern, Factorization using the Elliptic Curve Method.

Crossrefs

Cf. A033879, A063990, A104072, A204388, A275701 (abundance 26).

Programs

Magma

[n: n in [1..2*10^6] | (2*n-SumOfDivisors(n)) eq 26]; // Vincenzo Librandi, Aug 06 2016

Mathematica

Select[Range[10^6], 2 # - (DivisorSigma[1, #]) == 26 &] (* Vincenzo Librandi, Aug 06 2016 *)

PARI

is(n) = 2*n-sigma(n)==26 \\ Felix Fröhlich, Aug 06 2016

Extensions

a(9) from Giovanni Resta, Aug 07 2016

a(10) from Hiroaki Yamanouchi, Aug 21 2018