A073713 Numbers n such that the number of distinct primes dividing n = number of anti-divisors of n.

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

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