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A239875
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Friendly club of (first Wieferich prime)-1 and (second Wieferich prime)-1: numbers n such that sigma(n)/n = 112/39.
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1
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OFFSET
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1,1
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COMMENTS
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Dobson (see Link section) noted that A001220(1)-1 and A001220(2)-1 are friendly numbers, i.e., they have the same abundancy index (namely, 112/39), hence belong to the same "club" of friendly numbers.
Numbers n such that sigma(n)/n = 112/39 where sigma is A000203.
Clearly, all terms are multiples of 39. Note that 39 itself satisfies sigma(39)/39 = 56/39, half the desired value. Then if s is a perfect number (s is in A000396) and if s is coprime to 39, we have that 39*s belongs to the sequence, since sigma(39*s)/(39*s) = (sigma(39)/39)*(sigma(s)/s) = (56/39)*2 = 112/39. Michel Marcus made me aware of this fact. All even perfect numbers s other than s=6 satisfy the requirement, so this gives a lot of new terms. Other terms in this sequence (of the form 39 times perfect numbers) are thus: 335004893184, 5360108961792, 89927877317458132992, 103679783671223438041533012022199844864 etc.
The only known terms that do not come from 39 times a perfect number, are a(2)=3510 and a(5)=11504844. Their cofactors with 39 are not coprime to 39. Are there more numbers a(k) such that gcd(a(k)/39, 39) > 1?
How often is a(k)+1 a prime number? For k=1 and k=2 we get Wieferich primes. The number 39*A000396(8) belongs to the sequence, and 39*A000396(8)+1 is a prime (though not a Wieferich prime).
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LINKS
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EXAMPLE
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19344 is included because sigma(19344)/19344 = 55552/19344 = 112/39.
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PROG
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(PARI) forstep(n=39, 10^10, 39, if(sigma(n)*39==n*112, print1(n, ", ")))
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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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