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A073713
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Numbers n such that the number of distinct primes dividing n = number of anti-divisors of n.
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1
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1, 3, 4, 12, 24, 30, 36, 114, 120, 156, 174, 516, 576, 744, 804, 834, 894, 1056, 1344, 1356, 1626, 1686, 1884, 2064, 2136, 2274, 2616, 3396, 3414, 3606, 4044, 4146, 4314, 4506, 5034, 5136, 6036, 6054, 6126, 6306, 6504, 7296, 7680, 7824, 7944, 8994, 9024
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OFFSET
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1,2
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COMMENTS
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See A066272 for definition of anti-divisor.
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LINKS
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EXAMPLE
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30 is here since it has three distinct primes that divide it: {2, 3, 5} and three anti-divisors: {4, 12, 20}.
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MATHEMATICA
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atd[n_] := Count[Flatten[Quotient[#, Rest[Select[Divisors[#], OddQ]]] & /@ (2 n + Range[-1, 1])], Except[1]]; Select[Range[9030], PrimeNu[#] == atd[#] &] (* Jayanta Basu, Jul 08 2013 *)
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PROG
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(PARI) {for(n=1, 9050, v1=[]; v2=[]; v3=[]; ds=divisors(2*n-1); for(k=2, matsize(ds)[2]-1, if(ds[k]%2>0, v1=concat(v1, ds[k]))); ds=divisors(2*n); for(k=2, matsize(ds)[2]-1, if(ds[k]%2>0, v2=concat(v2, ds[k]))); ds=divisors(2*n+1); for(k=2, matsize(ds)[2]-1, if(ds[k]%2>0, v3=concat(v3, ds[k]))); v=vecsort(concat(v1, concat(v2, v3))); if(matsize(v)[2]==matsize(factor(n))[1], print1(n, ", ")))}
(Python3)
from sympy import divisors, factorint
A073713 = [n for n in range(1, 10**5) if len(factorint(n)) == len([2*d for d in divisors(n) if n > 2*d and n % (2*d)] + [d for d in divisors(2*n-1) if n > d >= 2 and n % d] + [d for d in divisors(2*n+1) if n > d >= 2 and n % d])] # Chai Wah Wu, Aug 13 2014
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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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