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A283024
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Natural numbers n such that the number of primes of the form d(n)/x is equal to the number of primes of the form sigma(n)/y where x, y are divisors of n.
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0
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1, 2, 4, 6, 9, 15, 16, 21, 24, 25, 27, 28, 30, 33, 35, 39, 42, 48, 51, 54, 55, 57, 64, 65, 66, 69, 70, 77, 78, 85, 87, 90, 91, 93, 95, 100, 102, 105, 110, 111, 112, 114, 115, 119, 120, 123, 125, 129, 130, 133, 135, 138, 141, 143, 145, 154, 155, 159, 161, 165
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OFFSET
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1,2
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COMMENTS
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In this sequence there are no odd prime numbers, but there is even prime number 2.
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LINKS
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EXAMPLE
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2 is in this sequence because d(2)/1 = 2 is prime and sigma(2)/1 = 3 is prime, d(2)/2 = 1 is no prime and sigma(2)/2 = 3/2 is no prime, where 1, 2 are divisors of 2.
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MATHEMATICA
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okQ[n_] := Block[{d = Divisors[n], t0, t1=DivisorSigma[1, n]}, t0 = Length@ d; Length @Select[d, PrimeQ[ t0/#] &] == Length@ Select[d, PrimeQ[ t1/#] &]]; Select[Range@ 1000, okQ] (* Giovanni Resta, Mar 05 2017 *)
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PROG
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(PARI) is(n)=my(f=factor(n), d=numdiv(f), fd=factor(d)[, 1], s=sigma(f), fs=factor(s)[, 1]); sum(i=1, #fd, n%(d/fd[i])==0)==sum(i=1, #fs, n%(s/fs[i])==0) \\ Charles R Greathouse IV, Feb 27 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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