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A275701
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Numbers n whose abundance is 26: sigma(n) - 2n = 26.
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4
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80, 1184, 6464, 29312, 78975, 510464, 557192, 137431875584, 549741658112, 8796036399104, 35184258842624, 2251798907715584
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Select[Range[10^7], DivisorSigma[1, #] - 2 # == 26 &] (* Vincenzo Librandi, Sep 16 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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