OFFSET
1,1
COMMENTS
From Enrique P�rez Herrero, Jul 06 2012: (Start)
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
T. D. Noe, Table of n, a(n) for n = 1..1000
Debmalya Basak, Nicolas Robles, and Alexandru Zaharescu, Exponential sums over M�bius 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 P�rez 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
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A030059(n):
def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b+1, isqrt(x//c)+1), a+1)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b+1, integer_nthroot(x//c, m)[0]+1), a+1) for d in g(x, a2, b2, c*b2, m-1)))
def f(x): return int(n+x-primepi(x)-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, i)) for i in range(3, x.bit_length(), 2)))
kmin, kmax = 0, 1
while f(kmax) > kmax:
kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax # Chai Wah Wu, Aug 29 2024
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved