A182358 Numbers n for which the number of divisors of n is congruent to 2 mod 4.

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 112, 113, 116, 117, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Author

Douglas Latimer, Apr 26 2012

Keywords

• nonn
• easy

Comments

The product of any 2 terms a(i)*a(j) is not a member of the sequence.

tau(n) is congruent to 2 modulo 4 iff only one prime in the prime factorization of n has exponent of the form 4*m + 1, and no prime in the prime factorization of n has exponent of the form 4*k + 3.

Examples

			The divisors of 12 are: 1, 2, 3, 4, 6, 12 [6 divisors]. 6 is congruent to 2 modulo 4. Thus 12 is a member of this sequence.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

This is an extension of A000040 (the prime numbers, which each have 2 divisors). The definition of this sequence uses A000005 (the number of divisors of n).

Programs

Mathematica

Select[Range[200],Mod[DivisorSigma[0,#],4]==2&] (* Harvey P. Dale, Sep 07 2020 *)

PARI

{plnt=1 ;  for(k=1, 10^7,
if(numdiv(k) % 4 == 2, print1(k, ", "); plnt++ ; if(100 <  plnt, break() )))}

PARI

is(n)=my(p=core(n));isprime(p)&&valuation(n,p)%4==1 \\ Charles R Greathouse IV, Apr 26 2012

PARI

list(lim)=my(v=List(),t);forprime(p=2,lim,for(m=1,sqrtint(lim\p), if(m%p==0, next); t=p*m^2; for(n=1,sqrtint(sqrtint(lim\t)), listput(v, t*n^4)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Apr 26 2012

Formula

Terms are of the form p * m^2 * n^4 for any prime p, m coprime to p, and n. - Charles R Greathouse IV, Apr 26 2012