

A030059


Numbers that are the product of an odd number of distinct primes.


19



2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 181, 182, 186, 190, 191, 193
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OFFSET

1,1


COMMENTS

From Enrique Pérez Herrero, Jul 06 2012: (Start)
This sequence and A030229 form a partition of the squarefree numbers set: A005117.
Also solutions to equation mu(n)=1.
Sum_{n>=1} 1/a(n)^s = (Zeta(s)^2  Zeta(2*s))/(2*Zeta(s)*Zeta(2*s)).
(End) [See A088245 and the Hardy reference.  Wolfdieter Lang, Oct 18 2016]
The lexicographically least sequence of integers > 1 such that for each entry, the number of proper divisors occurring in the sequence is equal to 0 modulo 3.  Masahiko Shin, Feb 12 2018


REFERENCES

B. C. Berndt & R. A. Rankin, Ramanujan: Letters and Commentary, see p. 23; AMS Providence RI 1995.
G. H. Hardy, Ramanujan, AMS Chelsea Publishing, 2002, pp. 64  65, (misprint on p. 65, line starting with Hence: it should be ... 1/Zeta(s) not ... Zeta(s)).
S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962, p. xxiv, 21.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105106.
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Moebius Function
Eric Weisstein's World of Mathematics, Prime Sums


FORMULA

omega(a(n)) = A001221(a(n)) gives A005408. {primes A000040} UNION {sphenic numbers A007304} UNION {numbers that are divisible by exactly 5 different primes A051270} UNION {products of 7 distinct primes (squarefree 7almost primes) A123321} UNION {products of 9 distinct primes; also n has exactly 9 distinct prime factors and n is squarefree A115343} UNION....  Jonathan Vos Post, Oct 19 2007
a(n) < n*Pi^2/3 infinitely often; a(n) > n*Pi^2/3 infinitely often.  Charles R Greathouse IV, Sep 07 2017


MAPLE

a := n > `if`(numtheory[mobius](n)=1, n, NULL); seq(a(i), i=1..193); # Peter Luschny, May 04 2009


MATHEMATICA

Select[Range[300], MoebiusMu[#] == 1 &] (* Enrique Pérez Herrero, Jul 06 2012 *)


PROG

(PARI) is(n)=my(f=factor(n)[, 2]); #f%2 && vecmax(f)==1 \\ Charles R Greathouse IV, Oct 16 2015
(PARI) is(n)=moebius(n)==1 \\ Charles R Greathouse IV, Jan 31 2017


CROSSREFS

Cf. A000040, A001221, A005408, A007304, A030231, A051270, A123321, A030229, A005117, A088245.
Sequence in context: A079603 A095959 A028905 * A201879 A319333 A089063
Adjacent sequences: A030056 A030057 A030058 * A030060 A030061 A030062


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from David W. Wilson


STATUS

approved



