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A030059 Numbers that are the product of an odd number of distinct primes. 57
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 partition the squarefree numbers: A005117.
Also solutions to the 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
The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, May 22 2020
Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = -n, where sigma(n) is the sum of divisors function (A000203). - Robert D. Rosales, May 20 2024
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
Debmalya Basak, Nicolas Robles, and Alexandru Zaharescu, Exponential sums over Möbius convolutions with applications to partitions, arXiv:2312.17435 [math.NT], 2023. Mentions this sequence.
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
H. S. Wilf, A Greeting; and a view of Riemann's Hypothesis, Amer. Math. Monthly, 94:1 (1987), 3-6.
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
# alternative
A030059 := proc(n)
option remember;
local a;
if n = 1 then
2;
else
for a from procname(n-1)+1 do
if numtheory[mobius](a) = -1 then
return a;
end if;
end do:
end if;
end proc: # R. J. Mathar, Sep 22 2020
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
Sequence in context: A359802 A028905 A365829 * A201879 A327783 A319333
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved

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Last modified July 12 23:13 EDT 2024. Contains 374257 sequences. (Running on oeis4.)