A283024 Natural numbers n such that the number of primes of the form d(n)/x is equal to the number of primes of the form sigma(n)/y where x, y are divisors of n.

1, 2, 4, 6, 9, 15, 16, 21, 24, 25, 27, 28, 30, 33, 35, 39, 42, 48, 51, 54, 55, 57, 64, 65, 66, 69, 70, 77, 78, 85, 87, 90, 91, 93, 95, 100, 102, 105, 110, 111, 112, 114, 115, 119, 120, 123, 125, 129, 130, 133, 135, 138, 141, 143, 145, 154, 155, 159, 161, 165

Offset

1,2

Comments

In this sequence there are no odd prime numbers, but there is even prime number 2.

Links

Table of n, a(n) for n=1..60.

Example

2 is in this sequence because d(2)/1 = 2 is prime and sigma(2)/1 = 3 is prime, d(2)/2 = 1 is no prime and sigma(2)/2 = 3/2 is no prime, where 1, 2 are divisors of 2.

Mathematica

okQ[n_] := Block[{d = Divisors[n], t0, t1=DivisorSigma[1, n]}, t0 = Length@ d; Length @Select[d, PrimeQ[ t0/#] &] == Length@ Select[d, PrimeQ[ t1/#] &]]; Select[Range@ 1000, okQ] (* Giovanni Resta, Mar 05 2017 *)

Prog

(PARI) is(n)=my(f=factor(n), d=numdiv(f), fd=factor(d)[, 1], s=sigma(f), fs=factor(s)[, 1]); sum(i=1, #fd, n%(d/fd[i])==0)==sum(i=1, #fs, n%(s/fs[i])==0) \\ Charles R Greathouse IV, Feb 27 2017

Crossrefs

Cf. A000005, A000203.

Sequence in context: A218605 A024849 A323227 * A090483 A299494 A306597

Adjacent sequences: A283021 A283022 A283023 * A283025 A283026 A283027

Keyword

nonn

Author

Juri-Stepan Gerasimov, Feb 27 2017

Extensions

a(7)-a(44) from Charles R Greathouse IV, Feb 27 2017

a(45)-a(60) from Giovanni Resta, Mar 05 2017

Status

approved