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A030059 Numbers that are the product of an odd number of distinct primes. 18
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 (list; graph; refs; listen; history; text; internal format)
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) 105-106.

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 7-almost 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 A089063 A247142

Adjacent sequences:  A030056 A030057 A030058 * A030060 A030061 A030062

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified February 20 02:47 EST 2018. Contains 299357 sequences. (Running on oeis4.)