

A267076


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


2



98671027044178875, 110707233329488965, 116112602631024285, 138000805229957475, 347701424633005443, 373158429641554557, 581028000695478105, 650656605928265895, 1057340336068271871, 1192941584025936129, 1317952177931347245, 1597858331524012755
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OFFSET

1,1


COMMENTS

Both members of the amicable pair with same sopfr are listed in the sequence but are not necessarily adjacent.
The terms shown have 17, 18 digits (7 terms) and 19 digits (four terms).
Sergei Chernykh has conducted several searches for amicable pairs in the 18digit range (here p and q are the largest prime factors):
1) All pairs of the form (m*p^k1, n*q^k2) where k1 > 1 _OR_ k2 > 1
2) All pairs of the form (m*p, n*q) where m < 2*10^11 _AND_ n < 2*10^11
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
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.
This means that all 18digit members of A267076 are already known. There are no new ones. (End)
Sergei Chernykh with BOINC completed the amicable pairs list with 20 digits.
In their ongoing search for 21digit 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


LINKS



EXAMPLE

To illustrate that 347701424633005443 and 373158429641554557 belong to the sequence:
347701424633005443 = 3*7^3*11*13*19*37*41*59*131*10607, sopfr(n) = 10942.
373158429641554557 = 3*7^2*11*13*19*311*383*839*9349, sopfr(n) = 10942.


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



