A030513 Numbers with 4 divisors.

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187

Offset

1,1

Comments

Essentially the same as A007422.

Numbers which are either the product of two distinct primes (A006881) or the cube of a prime (A030078).

4*a(n) are the solutions to A048272(x) = Sum_{d|x} (-1)^d = 4. - Benoit Cloitre, Apr 14 2002

Since A119479(4)=3, there are never more than 3 consecutive integers in the sequence. Triples of consecutive integers start at 33, 85, 93, 141, 201, ... (A039833). No such triple contains a term of the form p^3. - Ivan Neretin, Feb 08 2016

Numbers that are equal to the product of their proper divisors (A007956) (proof in Sierpiński). - Bernard Schott, Apr 04 2022

References

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Links

R. J. Mathar, Table of n, a(n) for n = 1..1000

R. J. Mathar, Maple programs for A030638, A030637, A030636, A030635, A030634, A030633, A030632, A030631, A030630, A030629, A030628, A030627, A030626, A030516, A030515, A030514, A030513

Formula

{n : A000005(n) = 4}. - Juri-Stepan Gerasimov, Oct 10 2009

Mathematica

Select[Range[200], DivisorSigma[0, #]==4&] (* Harvey P. Dale, Apr 06 2011 *)

Prog

(PARI) is(n)=numdiv(n)==4 \\ Charles R Greathouse IV, May 18 2015
(Magma) [n: n in [1..200] | DivisorSigma(0, n) eq 4]; // Vincenzo Librandi, Jul 16 2015
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A030513(n):
    def f(x): return int(n+x-primepi(integer_nthroot(x, 3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 16 2024

Crossrefs

Cf. A000005, A007422, A007956, A030515, A035533, A048272, A119479.

Equals the disjoint union of A006881 and A030078.

Sequence in context: A291127 A211337 A007422 * A161918 A294729 A242270

Adjacent sequences: A030510 A030511 A030512 * A030514 A030515 A030516

Keyword

nonn,easy,nice

Author

Jeff Burch

Extensions

Incorrect comments removed by Charles R Greathouse IV, Mar 18 2010

Status

approved