A104141 -id:A104141 - cp's OEIS frontend

A030229 Numbers that are the product of an even number of distinct primes.

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

Offset: 1

David W. Wilson

These are the positive integers k with moebius(k) = 1 (cf. A008683). - N. J. A. Sloane, May 18 2021
From Enrique Pérez Herrero, Jul 06 2012: (Start)
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)
A008683(a(n)) = 1; a(A220969(n)) mod 2 = 0; a(A220968(n)) mod 2 = 1. - Reinhard Zumkeller, Dec 27 2012
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
From Peter Munn, Oct 04 2019: (Start)
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

			(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,...

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.

T. D. Noe, Table of n, a(n) for n = 1..1000
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.

A006881, A046386, A067885, A123322, A281222 are subsequences.

Cf. A000379, A005117, A008683, A028260, A030059, A104141, A151797, A245630.

import Data.List (elemIndices)
a030229 n = a030229_list !! (n-1)
a030229_list = map (+ 1) $ elemIndices 1 a008683_list
-- Reinhard Zumkeller, Dec 27 2012

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
A030229 := proc(n)
    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:
seq(A030229(n),n=1..40) ; # R. J. Mathar, Sep 22 2020

Select[Range[214], MoebiusMu[#] == 1 &] (* Jean-François Alcover, Oct 04 2011 *)

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

is(n)=moebius(n)==1 \\ Charles R Greathouse IV, Jan 31 2017
for(n=1,500, isA030229(n)&print1(n",")) \\ M. F. Hasler

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A030229(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-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
    return kmax # Chai Wah Wu, Aug 29 2024

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

A030059 Numbers that are the product of an odd number of distinct primes.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

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.

T. D. Noe, Table of n, a(n) for n = 1..1000
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.

Cf. A000040, A001221, A005408, A007304, A008683, A030231, A051270, A123321, A030229, A005117, A088245, A104141.

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

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

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

is(n)=moebius(n)==-1 \\ Charles R Greathouse IV, Jan 31 2017

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

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

More terms from David W. Wilson

A003657 Discriminants of imaginary quadratic fields, negated.

Original entry on oeis.org

3, 4, 7, 8, 11, 15, 19, 20, 23, 24, 31, 35, 39, 40, 43, 47, 51, 52, 55, 56, 59, 67, 68, 71, 79, 83, 84, 87, 88, 91, 95, 103, 104, 107, 111, 115, 116, 119, 120, 123, 127, 131, 132, 136, 139, 143, 148, 151, 152, 155, 159, 163, 164, 167, 168, 179, 183, 184, 187, 191

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

Negative of fundamental discriminants D := b^2-4*a*c<0 of definite integer binary quadratic forms F=a*x^2+b*x*y+c*y^2. See Buell reference pp. 223-234. See 4*A089269 = A191483 for even a(n) and A039957 for odd a(n). - Wolfdieter Lang, Nov 07 2003
All prime numbers in the set of the absolute values of negative fundamental discriminants are Gaussian primes (A002145). - Paul Muljadi, Mar 29 2008
Complement: 1, 2, 5, 6, 9, 10, 12, 13, 14, 16, 17, 18, 21, 22, 25, 26, 27, 28, 29, 30, 32, 33, 34, 36, ..., . - Robert G. Wilson v, Jun 04 2011
The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, Feb 23 2021

Duncan A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989.
Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, p. 514.
Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..3000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See pp. 30, 53.
Steven R. Finch, Class number theory. [Cached copy, with permission of the author]
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Eric Weisstein's World of Mathematics, Class Number, Dirichlet L-Series, Fundamental Discriminant.

Cf. A002145, A003658, A039957 (odd terms), A191483 (even terms), A104141.

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric E. Weisstein *); -Select[-Range@ 194, FundamentalDiscriminantQ] (* Robert G. Wilson v, Jun 01 2011 *)

ok(n)={isfundamental(-n)} \\ Andrew Howroyd, Jul 20 2018

ok(n)={n<>1 && issquarefree(n/2^valuation(n,2)) && (n%4==3 || n%16==8 || n%16==4)} \\ Andrew Howroyd, Jul 20 2018

[n for n in (1..200) if is_fundamental_discriminant(-n)==1] # G. C. Greubel, Mar 01 2019

A003968 Möbius transform of A003959.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 18, 12, 10, 11, 18, 13, 14, 15, 54, 17, 24, 19, 30, 21, 22, 23, 54, 30, 26, 48, 42, 29, 30, 31, 162, 33, 34, 35, 72, 37, 38, 39, 90, 41, 42, 43, 66, 60, 46, 47, 162, 56, 60, 51, 78, 53, 96, 55, 126, 57, 58, 59, 90, 61, 62, 84, 486, 65, 66, 67, 102, 69

Offset: 1

Marc LeBrun

a(n) = n for squarefree n; otherwise, a(n) > n. - Ivan Neretin, May 13 2015

Dirichlet inverse of A062953. - Werner Schulte, Oct 25 2018

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A003959, A005596, A062953, A104141.

Table[{pp, aa} = Transpose[FactorInteger[n]]; Times @@ (pp*(pp + 1)^(aa - 1)), {n, 70}]  (* Ivan Neretin, May 13 2015 *)

a(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i,2]-1); f[i, 2] = 1); factorback(f);} \\ Michel Marcus, Feb 26 2015

Multiplicative with a(p^e) = p(p+1)^(e-1). - David W. Wilson, Sep 01 2001

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 + 1/(p^3 - p^2 - p)) = A104141/A005596 = 0.8128327996... . - Amiram Eldar, Oct 23 2022

More terms from David W. Wilson, Aug 29 2001

A276378 Numbers k such that 6*k is squarefree.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 203, 205, 209, 211

Offset: 1

Juri-Stepan Gerasimov, Sep 02 2016

These are the numbers from A005117 that are not divisible by 2 and 3.
Squarefree numbers coprime to 6. - Robert Israel, Sep 02 2016
Numbers k such that A008588(k) is in A005117. - Felix Fröhlich, Sep 02 2016
The asymptotic density of this sequence is 3/Pi^2 (A104141). - Amiram Eldar, May 22 2020
From Peter Munn, Nov 20 2020: (Start)
The products generated from each subset of A215848 (primes greater than 3).
Closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897. (End)
Multiplied by 6 we have 6, 30, 42, 66, 78, 102, ..., the values that may appear in A076978 after the 1, 2. [Don Reble, Dec 02 2020] - R. J. Mathar, Dec 15 2020
By the von Staudt-Clausen theorem, denominators of Bernoulli numbers are of the form 6*a(n) for some n. - Charles R Greathouse IV, May 16 2024

			5 is in this sequence because 6*5 = 30 = 2*3*5 is squarefree.

Robert Israel, Table of n, a(n) for n = 1..10000

Numbers m such that k*m is squarefree: A005117 (k = 1), A056911 (k = 2), A261034 (k = 3), A274546 (k = 5).
Subsequence of A007310, A300957, and A339690.
Cf. A003961, A008588, A059897, A076978, A104141, A215848.

[n: n in [1..230] | IsSquarefree(6*n)];

select(numtheory:-issqrfree, [seq(seq(6*i+j,j=[1,5]),i=0..100)]); # Robert Israel, Sep 02 2016

Select[Range@ 212, SquareFreeQ[6 #] &] (* Michael De Vlieger, Sep 02 2016 *)

is(n) = issquarefree(6*n) \\ Felix Fröhlich, Sep 02 2016

{a(n) : n >= 1} = {A003961(A003961(A005117(n))) : n >= 1} = {A003961(A056911(n)) : n >= 1}. - Peter Munn, Nov 20 2020

Sum_{n>=1} 1/a(n)^s = (6^s)*zeta(s)/((1+2^s)*(1+3^s)*zeta(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).

Original entry on oeis.org

1, 3, 2, 12, 96, 1152, 2304, 41472, 165888, 3981312, 119439360, 3822059520, 7644119040, 321052999680, 1284211998720, 61642175938560, 3328677500682240, 199720650040934400, 399441300081868800, 1597765200327475200, 115039094423578214400, 230078188847156428800, 18406255107772514304000

Offset: 1

Jonathan Sondow, Feb 01 2014

A236436(n)/(a(n)*zeta(2)) is the asymptotic density of the prime(n-1)-rough squarefree numbers (squarefree numbers whose prime factors are all >= prime(n-1)) for n >= 2. E.g., A236436(2)/(a(2)*zeta(2)) = 2/(3*zeta(2)) = 4/Pi^2 (A185199) is the asymptotic density of the odd squarefree numbers (A056911), and A236436(3)/(a(3)*zeta(2)) = 1/(2*zeta(2)) = 3/Pi^2 (A104141) is the asymptotic density of the 5-rough squarefree numbers (A276378). - Amiram Eldar, Aug 26 2025

			(1 + 1/2)*(1 + 1/3)*(1 + 1/5)*(1 + 1/7) = 96/35 has numerator a(5) = 96.
Fractions begin with 1, 3/2, 2, 12/5, 96/35, 1152/385, 2304/715, 41472/12155, 165888/46189, 3981312/1062347, 119439360/30808063, 3822059520/955049953, ...

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979; Theorem 429.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Jonathan Sondow and Eric W. Weisstein, MathWorld: Mertens Theorem.

Cf. A038110, A060753, A072044, A072045, A073004, A236436 (denominators), A335004.

Cf. A013661, A056911, A104141, A185199, A276378.

Numerator@Table[ Product[ 1 + 1/Prime[ k], {k, 1, n-1}], {n, 1, 23}]

a(n+1) / A236436(n+1) = (A072045(n)/A072044(n)) / (A038110(n+1)/A060753(n+1)) because 1+x = (1-x^2) / (1-x).

a(n) / A236436(n) = Product_{k=1..n-1} (1 + 1/prime(k)) ~ (6/Pi^2)*exp(gamma)*log(n) as n -> infinity, by Mertens's theorem.

A064018 a(n) = A002088(10^n) = Sum_{k <= 10^n} phi(k), sum of the Euler totients phi = A000010.

Original entry on oeis.org

1, 32, 3044, 304192, 30397486, 3039650754, 303963552392, 30396356427242, 3039635516365908, 303963551173008414, 30396355092886216366, 3039635509283386211140, 303963550927059804025910, 30396355092702898919527444, 3039635509270144893910357854, 303963550927013509478708835152

Offset: 0

Robert G. Wilson v, Sep 07 2001

nonn

Asymptotically, A002088(n) ~ 0.30396355...*n^2 = (3/Pi^2)*n^2, see A104141 and A002088. - Michael B. Porter, Mar 08 2013 [corrected by M. F. Hasler, Apr 18 2015]

			a(1) = phi(1) + ... + phi(10) = 1 + 1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 + 4 = 32.

Lucas A. Brown, Table of n, a(n) for n = 0..19 (terms 0..18 from Hiroaki Yamanouchi)
Lucas A. Brown, Python program.
Lucas Augustus Brown, Computation of the Totient Summatory Function, arXiv:2506.07386 [math.NT], 2025.
Eric Weisstein's World of Mathematics, Totient Summatory Function.
Wikipedia, Totient summatory function.

Cf. A000010, A002088, A104141.

s = 0; k = 1; Do[ While[ k <= 10^n, s = s + EulerPhi[ k ]; k++ ]; Print[ s ], {n, 0, 8} ]

# See LINKS. - Lucas A. Brown, Jun 08 2025

a(n) = Sum_{k <= 10^n} A000010(k).

More terms from Robert G. Wilson v, Sep 07 2001
a(10)-a(11) from Donovan Johnson, Feb 06 2010
a(12) from Donovan Johnson, Feb 07 2012
a(13)-a(14) from Hiroaki Yamanouchi, Jul 06 2014
a(15) from Asif Ahmed, Apr 16 2015
Name edited by Michel Marcus and M. F. Hasler, Apr 16 and Apr 18 2015

A262228 Deficiency sequence: a(0) = 1, a(n) is the smallest prime p > a(n-1) such that the product of a(i), 1 <= i < n, is deficient (belongs to A005100).

Original entry on oeis.org

1, 2, 5, 11, 59, 653, 84761, 2763189059, 377406001499268899, 2638619515495963542360422694651593, 135435890329895562961039215198033899386421965445591860752412324961

Offset: 0

Chayim Lowen, Sep 15 2015

nonn

The product of the first n+1 terms is the smallest deficient multiple of the product of the first n terms.
The product of any finite number of distinct terms of this sequence is deficient.
a(n) for n > 0 is the lexicographically earliest sequence of primes P, such that the asymptotic density of the squarefree numbers (A005117) which are not divisible by any prime in P is 3/Pi^2 (A104141), i.e., half the asymptotic density of all the squarefree numbers. - Amiram Eldar, Nov 30 2020

			a(3) = 11 because A001065(2*5*7) = A001065(70) = 74 > 70, and A001065(2*5*11) = A001065(110) = 106 < 110.
From _M. F. Hasler_, Dec 14 2017: (Start)
Let Q(x) = 1/(2x/sigma(x) - 1), P(n) = Product( a(k), k<n): P(0) = 1 (empty product). Then:
Q(P(0)) = 1, a(0) = nextprime(1) = 2 = P(1).
Q(P(1)) = 3, a(1) = 5. (2*3 is perfect, P(2) = 2*5 is deficient.)
Q(P(2)) = 9, a(2) = 11. (2*5*7 is weird, P(3) = 2*5*11 is deficient.)
Q(P(3)) = 54, a(3) = 59. (P(3)*53 is weird, P(4) = 2*5*11*59 is deficient.)
Q(P(4)) = 648, a(4) = 653. (P(4)*647 is weird, P(5) = 2*5*11*59*653 is deficient.)
Q(P(5)) = 84758.4, a(5) = 84761. (P(5)*84751 is abundant and semiperfect: sum of all proper divisors except {1, 2, 11, 22, 55, 59, 590}; P(6) = 2*5*11*59*653*84761 is deficient.) (End)

Amiram Eldar, Table of n, a(n) for n = 0..14

Cf. A001065, A005100, A005117, A104141, A151800 (nextprime).

Cf. A002975 (primitive weird numbers), A000203 (sigma), A295001 (same definition but a(0) = 4).

a[0]=1; a[n_] := a[n] = NextPrime[1/(2*Product[a[i],{i,1,n-1}]/Product[a[i]+1,{i,1,n-1}]-1)]; Array[a, 11, 0] (* Amiram Eldar, Jun 10 2019 *)

lista(nn) = {print1(p=1, ", "); vp = [p]; for (n=2, nn, np = nextprime(1+floor(1/(2*prod(i=2, n-1, vp[i]/(vp[i]+1))-1))); vp = concat(vp, np); print1(np, ", "););} \\ Michel Marcus, Oct 16 2015

a=List(); m=1; for(n=0, 13, listput(a, p=nextprime(1\(2/sigma(m,-1)-1)+1)); p>default(primelimit)&&addprimes(p); m*=p); a \\ M. F. Hasler, Dec 14 2017

a(n) = A151800(floor(1/(2*(Product_{i=2..n-1} a(i)/(a(i)+1))-1))), where A151800 is the "next larger prime" function.
Lim_{n->infinity} A001065(Product_{i=0..n} a(i))/(Product_{i=0..n} a(i)) = 1. [Corrected by M. F. Hasler, Dec 04 2017]
Conjecture: log(a(n)) ~ e^(an+b) where a and b are approximately 0.6 and -1.6 respectively.

A070549 a(n) = Cardinality{ k in range 1 <= k <= n such that Moebius(k) = -1 }.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 21, 22, 22, 22, 23, 24, 24, 25, 25, 25, 25, 25, 26

Offset: 1

Benoit Cloitre, May 02 2002

mu(k)=-1 if k is the product of an odd number of distinct primes. See A057627 for mu(k)=0.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A008683, A057627.
Partial sums of A252233.
Cf. A002321, A013928, A030059, A070548, A104141.

ListTools:-PartialSums([seq(-min(numtheory:-mobius(n),0),n=1..100)]); # Robert Israel, Jan 08 2018

a[n_]:=Sum[Boole[MoebiusMu[k]==-1],{k,n}]; Array[a,78] (* Stefano Spezia, Jan 30 2023 *)

for(n=1,150,print1(sum(i=1,n,if(moebius(i)+1,0,1)),","))

From Amiram Eldar, Oct 01 2023: (Start)
a(n) = (A013928(n+1) - A002321(n))/2.
a(n) = A013928(n+1) - A070548(n).
a(n) = A070548(n) - A002321(n).
a(n) ~ (3/Pi^2) * n. (End)

A088246 Decimal expansion of 21/(2*Pi^2).

Original entry on oeis.org

1, 0, 6, 3, 8, 7, 2, 4, 2, 8, 2, 4, 4, 5, 4, 6, 6, 0, 0, 1, 6, 0, 7, 3, 4, 3, 6, 3, 7, 0, 2, 1, 4, 0, 2, 0, 8, 4, 9, 5, 7, 6, 7, 1, 3, 4, 0, 5, 8, 5, 8, 8, 7, 6, 2, 8, 7, 8, 8, 9, 4, 8, 3, 4, 8, 8, 8, 8, 1, 7, 7, 7, 0, 1, 0, 3, 4, 7, 2, 1, 3, 2, 5, 0, 7, 6, 9, 7, 3, 7, 6, 2, 1, 9, 0, 2, 9, 2, 9, 4, 9, 4

Offset: 1

Eric W. Weisstein, Sep 25 2003

			1.06387242824454660016073436370214020849576713405858...

Eric Weisstein's World of Mathematics, Moebius Function.
Index entries for transcendental numbers.

Cf. A007434, A030229, A082020, A088245.

Cf. A104141.

RealDigits[21/2/Pi^2, 10, 100][[1]] (* Amiram Eldar, May 23 2020 *)

21/2/Pi^2 \\ Charles R Greathouse IV, Oct 01 2022

Equals Sum_{k>=1} 1/A030229(k)^2. - Amiram Eldar, May 23 2020
Equals zeta(4)/zeta(6). - Terry D. Grant, Nov 07 2020
Equals Sum_{k>=1} A007434(k)/k^6. - Amiram Eldar, Jan 25 2024

cp's OEIS Frontend

A030229 Numbers that are the product of an even number of distinct primes.

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A030059 Numbers that are the product of an odd number of distinct primes.

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A003657 Discriminants of imaginary quadratic fields, negated.

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A003968 Möbius transform of A003959.

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A276378 Numbers k such that 6*k is squarefree.

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A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).

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