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

1,1

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.

Links

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

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

Example

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.

Mathematica

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

Prog

(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

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).

Sequence in context: A340657 A316793 A119251 * A336418 A212166 A293511

Adjacent sequences: A182355 A182356 A182357 * A182359 A182360 A182361

Keyword

nonn,easy

Author

Douglas Latimer, Apr 26 2012

Status

approved