A073713 - OEIS

%I #14 Dec 15 2017 17:35:59

%S 1,3,4,12,24,30,36,114,120,156,174,516,576,744,804,834,894,1056,1344,

%T 1356,1626,1686,1884,2064,2136,2274,2616,3396,3414,3606,4044,4146,

%U 4314,4506,5034,5136,6036,6054,6126,6306,6504,7296,7680,7824,7944,8994,9024

%N Numbers n such that the number of distinct primes dividing n = number of anti-divisors of n.

%C See A066272 for definition of anti-divisor.

%H Chai Wah Wu, <a href="/A073713/b073713.txt">Table of n, a(n) for n = 1..1000</a>

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

%t 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 *)

%o (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,",")))}

%o (Python3)

%o from sympy import divisors, factorint

%o 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

%Y Cf. A001221, A066272.

%K nonn

%O 1,2

%A _Jason Earls_, Aug 30 2002

%E Edited and extended by _Klaus Brockhaus_, Sep 02 2002