A030229 - OEIS (original) (raw)

1, 6, 10, 14, 15, 21, 22, 26, 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, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 210, 213, 214

COMMENTS

These are the positive integers k with moebius(k) = 1 (cf. A008683). - N. J. A. Sloane, May 18 2021

This sequence and A030059 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)

Characteristic function for values of a(n) = (mu(n)+1)! - 1, where mu(n) is the Mobius function (A008683). - Wesley Ivan Hurt, Oct 11 2013

Conjecture: For the matrix M(i,j) = 1 if j|i and 0 otherwise, Inverse(M)(a,1) = -1, for any a in this sequence. - Benedict W. J. Irwin, Jul 26 2016

Solutions to the equation Sum_{d|n} mu(d)*d = Sum_{d|n} mu(n/d)*d. - Torlach Rush, Jan 13 2018

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

Numbers n such that omega(n) = bigomega(n) = 2*k for some integer k.

The squarefree numbers in A000379.

The squarefree numbers in A028260.

This sequence is closed with respect to the commutative binary operation A059897(.,.), thus it forms a subgroup of the positive integers under A059897(.,.). A006094 lists a minimal set of generators for this subgroup. The lexicographically earliest ordered minimal set of generators is A100484 with its initial 4 removed.

(End)

The asymptotic density of this sequence is 3/Pi^2 (cf. A104141). - Amiram Eldar, May 22 2020

REFERENCES

B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 23; AMS Providence RI 1995

S. Ramanujan, Collected Papers, pp. xxiv, 21.

FORMULA

a(n) < n*Pi^2/3 infinitely often; a(n) > n*Pi^2/3 infinitely often. - Charles R Greathouse IV, Oct 04 2011; corrected Sep 07 2017

{a(n)} = {m : m = A059897(A030059(k), p), k >= 1} for prime p, where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Oct 04 2019

EXAMPLE

(empty product), 2*3, 2*5, 2*7, 3*5, 3*7, 2*11, 2*13, 3*11, 2*17, 5*7, 2*19, 3*13, 2*23,...

MAPLE

a := n -> `if`(numtheory[mobius](n)=1, n, NULL); seq(a(i), i=1..214); # Peter Luschny, May 04 2009

with(numtheory); t := [ ]: f := [ ]: for n from 1 to 250 do if mobius(n) = 1 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # Wesley Ivan Hurt, Oct 11 2013

# alternative

option remember;

local a;

if n = 1 then

1;

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:

PROG

(PARI) isA030229(n)= #(n=factor(n)[, 2]) % 2 == 0 && (!n || vecmax(n)==1 )

for(n=1, 500, isA030229(n)&print1(n", ")) \\ M. F. Hasler

(Haskell)

import Data.List (elemIndices)

a030229 n = a030229_list !! (n-1)

a030229_list = map (+ 1) $ elemIndices 1 a008683_list

(Python)

from math import isqrt, prod

from sympy import primerange, integer_nthroot, primepi

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-1+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(2, 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