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A275701 Numbers n whose abundance is 26: sigma(n) - 2n = 26. 4
80, 1184, 6464, 29312, 78975, 510464, 557192, 137431875584, 549741658112, 8796036399104, 35184258842624, 2251798907715584 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Any term x = a(m) can be combined with any term y = A275702(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: (x,y) = (1184,1210) = (a(2),A275702(5)) = (A063990(3),A063990(4)). If more are ever found, then they will also exhibit y-x = 26.

Notice that:

a(1) =     80 =   5* 16 = (2*4^2-27)*(4^2)

a(2) =   1184 =  37* 32 =   (4^3-27)*(4^3)/2

a(3) =   6464 = 101* 64 = (2*4^3-27)*(4^3)

a(4) =  29312 = 229*128 =   (4^4-27)*(4^4)/2

a(6) = 510464 = 997*512 =   (4^5-27)*(4^5)/2.

If p = 2*4^k-27 is prime and n = p*(p+27)/2, then it is not hard to show that sigma(n) - 2*n = 26. The values of k in A275767 will guarantee that p is prime (A275749). Similarly, if q = 4^k-27 is prime and n = q*(q+27)/2, then sigma(n) - 2*n = 26. The values of k in A274519 will guarantee that q is prime (A275750). So, the following values will be in this sequence and provide upper bounds for the next eight terms:

(2*4^9-27)*(4^9)     = 137431875584 >= a(8)

  (4^10-27)*(4^10)/2 = 549741658112 >= a(9)

  (4^11-27)*(4^11)/2 = 8796036399104 >= a(10)

(2*4^11-27)*(4^11)   = 35184258842624 >= a(11)

  (4^13-27)*(4^13)/2 = 2251798907715584 >= a(12)

  (4^25-27)*(4^25)/2 = 633825300114099501099609227264 >= a(13)

  (4^28-27)*(4^28)/2 = 2596148429267412841487728652582912 >= a(14)

  (4^29-27)*(4^29)/2 = 41538374868278617137133892585652224 >= a(15).

a(8) > 10^9. - Michel Marcus, Sep 15 2016

a(8) > 2*10^9. - Michel Marcus, Dec 31 2016

a(13) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018

LINKS

Table of n, a(n) for n=1..12.

D. Alpern, Factorization using the Elliptic Curve Method.

EXAMPLE

a(1) = 80, since sigma(80)-2*80 = 186-160 = 26.

a(2) = 1184, since sigma(1184)-2*1184 = 2394-2368 = 26.

a(3) = 6464, since sigma(6464)-2*6464 = 12954-12928 = 26.

MATHEMATICA

Select[Range[10^7], DivisorSigma[1, #] - 2 # == 26 &] (* Vincenzo Librandi, Sep 16 2016 *)

PROG

(PARI) isok(n) = sigma(n) - 2*n == 26; \\ Michel Marcus, Sep 15 2016

(MAGMA) [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 26]; // Vincenzo Librandi, Sep 16 2016

CROSSREFS

Cf. A033880, A063990, A274519, A275702 (deficiency 26), A275749, A275750, A275767.

Cf. A223609 (abundance 10), ..., A223613 (abundance 24).

Sequence in context: A264129 A223251 A251427 * A168364 A296353 A126861

Adjacent sequences:  A275698 A275699 A275700 * A275702 A275703 A275704

KEYWORD

nonn,more

AUTHOR

Timothy L. Tiffin, Aug 05 2016

EXTENSIONS

a(8)-a(12) from Hiroaki Yamanouchi, Aug 23 2018

STATUS

approved

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Last modified August 10 07:37 EDT 2020. Contains 336368 sequences. (Running on oeis4.)