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A261657
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Numbers with 3 prime factors a < b < c such that 2c = a^4 + b^2.
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3
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795, 3333, 11795, 20515, 25805, 38845, 40107, 48453, 66355, 106285, 108363, 183673, 184445, 236365, 265955, 329063, 347883, 605635, 856595, 1005715, 1068267, 1307047, 1356035, 1901485, 1955787, 2469379, 2733565, 3229795, 3571867
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OFFSET
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1,1
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COMMENTS
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This sequence is interesting as a less constrained form of A261656, or as a 3-prime-factored form of A261658.
Unlike A261656, this sequence has many examples of the b and c of one member being the a and b of another. Two such members of this sequence are 3333=3*11*101 and 13799731 = 11*101*12421. This lets us consider a four factor composite number using both, giving 3*11*101*12421 = 41399193. The sequence of four-factor composites such as this is A261658.
It would be nice to know if, in general, analogous n-factor composites exist.
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LINKS
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EXAMPLE
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The prime factors of 795 are 3, 5, and 53. 3^4=81, 5^2=25, and the average of 81 and 25 gives 53. Thus, 795 is a member.
The prime factors of 3333 are 3, 11, and 101. 3^4=81, 11^2=121, and the average of 81 and 121 gives 101. Thus, 3333 is a member.
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MAPLE
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n := 20: L := []: for a from 3 to n do if isprime(a) then for b from a to n^2 do if isprime(b) then c := (a^4+b^2)*(1/2); if isprime(c) then L := [op(L), a*b*c] end if end if end do end if end do; L := sort(L): L := remove(proc (t) options operator, arrow; (3/2)*n^2*(n^4+9) < t end proc, L)
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PROG
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(PARI) list(lim)=my(v=List(), t); forprime(b=5, , if(3*b*(b^2+81)/2>lim, break); forprime(a=3, b-2, my(c=(a^4+b^2)/2, t=a*b*c); if(t>lim, break); if(isprime(c), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Aug 29 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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