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A073713 Numbers n such that the number of distinct primes dividing n = number of anti-divisors of n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A066272 for definition of anti-divisor.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1000

EXAMPLE

30 is here since it has three distinct primes that divide it: {2, 3, 5} and three anti-divisors: {4, 12, 20}.

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A001221, A066272.

Sequence in context: A111358 A111357 A081621 * A291023 A084921 A070765

Adjacent sequences:  A073710 A073711 A073712 * A073714 A073715 A073716

KEYWORD

nonn

AUTHOR

Jason Earls, Aug 30 2002

EXTENSIONS

Edited and extended by Klaus Brockhaus, Sep 02 2002

STATUS

approved

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Last modified July 10 00:25 EDT 2020. Contains 335570 sequences. (Running on oeis4.)