A267076 - OEIS

%I #74 Feb 28 2021 00:47:44

%S 98671027044178875,110707233329488965,116112602631024285,

%T 138000805229957475,347701424633005443,373158429641554557,

%U 581028000695478105,650656605928265895,1057340336068271871,1192941584025936129,1317952177931347245,1597858331524012755

%N Amicable numbers with property that both members of the amicable pair have the same sopfr (as defined in A001414).

%C Both members of the amicable pair with same sopfr are listed in the sequence but are not necessarily adjacent.

%C The terms shown have 17, 18 digits (7 terms) and 19 digits (four terms).

%C Comment from _N. J. A. Sloane_, Jun 07 2016: (Start)

%C _Sergei Chernykh_ has conducted several searches for amicable pairs in the 18-digit range (here p and q are the largest prime factors):

%C 1) All pairs of the form (m*p^k1, n*q^k2) where k1 > 1 _OR_ k2 > 1

%C 2) All pairs of the form (m*p, n*q) where m < 2*10^11 _AND_ n < 2*10^11

%C 3) Current exhaustive search has already found all pairs of the form (m*p^k1, n*q^k2) where p < 21818622 for any k1, q, k2

%C If we combine the results of these searches it is easy to see that the remaining undiscovered pairs can only have the form (m*p, n*q) where their largest prime factors are p > 21818622 and q < 10000000 (2*10^18 / 2*10^11), so they can't have the same sopfr.

%C This means that all 18-digit members of A267076 are already known. There are no new ones. (End)

%C _Sergei Chernykh_ with BOINC completed the amicable pairs list with 20 digits.

%C In their ongoing search for 21-digit amicable numbers  _Sergei Chernykh_ and BOINC have so far found the following numbers: 130292188156891334007, 137813613144174393993, 208010335478545813941, 220018224493331050059, 250217395764910459875, 271313659794405815325, 276109509594435349833, 349735430520058090167, 370496519153268119073, 402333253352868456927, 781727874026691579075, 886084603302962180925. - _Sven Simon_ Feb 26 2021

%H Sven Simon, <a href="/A267076/b267076.txt">Table of n, a(n) for n = 1..22</a>

%H Sergei Chernykh, <a href="http://sech.me/ap/index.html">Amicable pairs list</a>

%e To illustrate that 347701424633005443 and 373158429641554557 belong to the sequence:

%e 347701424633005443 = 3*7^3*11*13*19*37*41*59*131*10607, sopfr(n) = 10942.

%e 373158429641554557 = 3*7^2*11*13*19*311*383*839*9349, sopfr(n) = 10942.

%Y Cf. A001414, A063990, A091340, A259180.

%K nonn

%O 1,1

%A _Sven Simon_, Jan 10 2016

%E Entry revised by _Michel Marcus_ and _N. J. A. Sloane_, Jan 23 2016