A083399 -id:A083399 - cp's OEIS frontend

A341444 Dirichlet inverse of A083399, where A083399(n) = 1 + omega(n).

1, -2, -2, 2, -2, 5, -2, -2, 2, 5, -2, -7, -2, 5, 5, 2, -2, -7, -2, -7, 5, 5, -2, 9, 2, 5, -2, -7, -2, -16, -2, -2, 5, 5, 5, 14, -2, 5, 5, 9, -2, -16, -2, -7, -7, 5, -2, -11, 2, -7, 5, -7, -2, 9, 5, 9, 5, 5, -2, 30, -2, 5, -7, 2, 5, -16, -2, -7, 5, -16, -2, -23, -2, 5, -7, -7, 5, -16, -2, -11, 2, 5, -2, 30, 5, 5, 5, 9, -2, 30, 5

Offset: 1

Michel Marcus, Feb 12 2021

sign

The Dirichlet inverse function, a(n) = (omega + 1)^(-1)(n). - Original name.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Carl-Erik Fröberg, On the prime zeta function, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187-202.
H. Hwang and S. Janson, A central limit theorem for random ordered factorizations of integers, Electron. J. Probab., 16(12):347-361, 2011.
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
M. D. Schmidt, New characterizations of the summatory function of the Moebius function, arXiv:2102.05842 [math.NT], 2021.
Index entries for sequences computed from exponents in factorization of n

Dirichlet inverse of A083399.
Cf. A001221, A001222, A008480, A008683, A008966, A341472 (partial sums).
Cf. also A334743.

a[1] = 1; a[n_] := a[n] = -DivisorSum[n, (PrimeNu[n/#] + 1)*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

cOmega(n) = if (n==1, 1, my(f=factor(n)); bigomega(n)!*prod(k=1, #f~, 1/f[k,2]!)); \\ A008480
a(n) = (-1)^bigomega(n)*sumdiv(n, d, moebius(n/d)^2*cOmega(d));

memoA341444 = Map();
A341444(n) = if(1==n,1,my(v); if(mapisdefined(memoA341444,n,&v), v, v = -sumdiv(n,d,if(dA341444(d),0)); mapput(memoA341444,n,v); (v))); \\ Antti Karttunen, Jul 21 2022~

a(n) = (-1)^A001222(n)*Sum_{d | n} A008683(n/d)^2*A008480(d).

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083399(n/d) * a(d). - Antti Karttunen, Jul 21 2022

Data section extended up to a(91) and name edited by Antti Karttunen, Jul 21 2022

A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244

Offset: 1

Eric W. Weisstein

nonn

Sometimes called "triprimes" or "3-almost primes".
See also A001358 for product of two primes (sometimes called semiprimes).
If you graph a(n)/n for n up to 10000 (and probably quite a bit higher), it appears to be converging to something near 3.9. In fact the limit is infinite. - Franklin T. Adams-Watters, Sep 20 2006
Meng shows that for any sufficiently large odd integer n, the equation n = a + b + c has solutions where each of a, b, c is 3-almost prime. The number of such solutions is (log log n)^6/(16 (log n)^3)*n^2*s(n)*(1 + O(1/log log n)), where s(n) = Sum_{q >= 1} Sum_{a = 1..q, (a, q) = 1} exp(i*2*Pi*n*a/q)*mu(n)/phi(n)^3 > 1/2. - Jonathan Vos Post, Sep 16 2005, corrected & rewritten by M. F. Hasler, Apr 24 2019
Also, a(n) are the numbers such that exactly half of their divisors are composite. For the numbers in which exactly half of the divisors are prime, see A167171. - Ivan Neretin, Jan 12 2016

			From _Gus Wiseman_, Nov 04 2020: (Start)
Also Heinz numbers of integer partitions into three parts, counted by A001399(n-3) = A069905(n) with ordered version A000217, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence of terms together with their prime indices begins:
      8: {1,1,1}     70: {1,3,4}     130: {1,3,6}
     12: {1,1,2}     75: {2,3,3}     138: {1,2,9}
     18: {1,2,2}     76: {1,1,8}     147: {2,4,4}
     20: {1,1,3}     78: {1,2,6}     148: {1,1,12}
     27: {2,2,2}     92: {1,1,9}     153: {2,2,7}
     28: {1,1,4}     98: {1,4,4}     154: {1,4,5}
     30: {1,2,3}     99: {2,2,5}     164: {1,1,13}
     42: {1,2,4}    102: {1,2,7}     165: {2,3,5}
     44: {1,1,5}    105: {2,3,4}     170: {1,3,7}
     45: {2,2,3}    110: {1,3,5}     171: {2,2,8}
     50: {1,3,3}    114: {1,2,8}     172: {1,1,14}
     52: {1,1,6}    116: {1,1,10}    174: {1,2,10}
     63: {2,2,4}    117: {2,2,6}     175: {3,3,4}
     66: {1,2,5}    124: {1,1,11}    182: {1,4,6}
     68: {1,1,7}    125: {3,3,3}     186: {1,2,11}
(End)

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974). See p. 211.

T. D. Noe, Table of n, a(n) for n = 1..10000
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909. See Vol. 1, p. 211.
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.
Eric Weisstein's World of Mathematics, Almost Prime

Cf. A000040, A001358 (biprimes), A014613 (quadruprimes), A033942, A086062, A098238, A123072, A123073, A101605 (characteristic function).
Cf. A109251 (number of 3-almost primes <= 10^n).
Subsequence of A145784. - Reinhard Zumkeller, Oct 19 2008
Cf. A007304 is the squarefree case.
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), this sequence (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
Cf. A253721 (final digits).
A014311 is a different ranking of ordered triples, with strict case A337453.
A046316 is the restriction to odds, with strict case A307534.
A075818 is the restriction to evens, with strict case A075819.
A285508 is the nonsquarefree case.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
Cf. A000212, A000217, A046389, A140106, A307719, A321773.

a014612 n = a014612_list !! (n-1)
a014612_list = filter ((== 3) . a001222) [1..] -- Reinhard Zumkeller, Apr 02 2012

with(numtheory); A014612:=n->`if`(bigomega(n)=3, n, NULL); seq(A014612(n), n=1..250) # Wesley Ivan Hurt, Feb 05 2014

threeAlmostPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@244, threeAlmostPrimeQ[ # ] &] (* Robert G. Wilson v, Jan 04 2006 *)
NextkAlmostPrime[n_, k_: 2, m_: 1] := Block[{c = 0, sgn = Sign[m]}, kap = n + sgn; While[c < Abs[m], While[ PrimeOmega[kap] != k, If[sgn < 0, kap--, kap++]]; If[ sgn < 0, kap--, kap++]; c++]; kap + If[sgn < 0, 1, -1]]; NestList[NextkAlmostPrime[#, 3] &, 2^3, 56] (* Robert G. Wilson v, Jan 27 2013 *)
Select[Range[244], PrimeOmega[#] == 3 &] (* Jayanta Basu, Jul 01 2013 *)

isA014612(n)=bigomega(n)==3 \\ Charles R Greathouse IV, May 07 2011

list(lim)=my(v=List(),t);forprime(p=2,lim\4, forprime(q=2,min(lim\(2*p),p), t=p*q; forprime(r=2,min(lim\t,q),listput(v,t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 04 2013

from sympy import factorint
def ok(n): f = factorint(n); return sum(f[p] for p in f) == 3
print(list(filter(ok, range(245)))) # Michael S. Branicky, Aug 12 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A014612(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 17 2024

def primeFactors(number: Int, list: List[Int] = List())
                                                      : List[Int] = {
  for (n <- 2 to number if (number % n == 0)) {
    return primeFactors(number / n, list :+ n)
  }
  list
}
(1 to 250).filter(primeFactors().size == 3) // _Alonso del Arte, Nov 04 2020, based on algorithm by Victor Farcic (vfarcic)

Product p_i^e_i with Sum e_i = 3.
a(n) ~ 2n log n / (log log n)^2 as n -> infinity [Landau, p. 211].
Tau(a(n)) = 2 * (omega(a(n)) + 1) = 2*A083399(a(n)), where tau = A000005 and omega = A001221. - Wesley Ivan Hurt, Jun 28 2013
a(n) = A078840(3,n). - R. J. Mathar, Jan 30 2019

More terms from Patrick De Geest, Jun 15 1998

A305614 Expansion of Sum_{p prime} x^p/(1 + x^p).

Original entry on oeis.org

0, 0, 1, 1, -1, 1, 0, 1, -1, 1, 0, 1, -2, 1, 0, 2, -1, 1, 0, 1, -2, 2, 0, 1, -2, 1, 0, 1, -2, 1, -1, 1, -1, 2, 0, 2, -2, 1, 0, 2, -2, 1, -1, 1, -2, 2, 0, 1, -2, 1, 0, 2, -2, 1, 0, 2, -2, 2, 0, 1, -3, 1, 0, 2, -1, 2, -1, 1, -2, 2, -1, 1, -2, 1, 0, 2, -2, 2, -1

Offset: 0

Gus Wiseman, Jun 06 2018

sign

a(n) is the number of prime divisors p|n such that n/p is odd, minus the number of prime divisors p|n such that n/p is even.

			The prime divisors of 12 are 2, 3. We see that 12/2 = 6, 12/3 = 4. None of those are odd, but both of them are even, so a(12) = -2.
The prime divisors of 30 are {2,3,5} with quotients {15,10,6}. One of these is odd and two are even, so a(30) = 1 - 2 = -1.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000005, A000607, A001221, A008683, A048165, A083399, A088705, A106404, A205745, A280954.

a:= n-> -add((-1)^(n/i[1]), i=ifactors(n)[2]):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 07 2018
# Alternative
N:= 1000: # to get a(0)..a(N)
V:= Vector(N):
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  R:= [seq(i,i=p..N,p)];
  W:= ;
  V[R]:= V[R]+W;
od:
[0,seq(V[i],i=1..N)]; # Robert Israel, Jun 07 2018

Table[Sum[If[PrimeQ[d], (-1)^(n/d - 1), 0], {d, Divisors[n]}], {n, 30}]

a(n) = -Sum_{p|n prime} (-1)^(n/p).
From Robert Israel, Jun 07 2018: (Start)
If n is odd, a(n) = A001221(n).
If n == 2 (mod 4), a(n) = 2 - A001221(n).
If n == 0 (mod 4) and n > 0, a(n) = -A001221(n). (End)
L.g.f.: log(Product_{k>=1} (1 + x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018

A048165 Expansion of Product_{k > 0} 1/(1 + x^prime(k)).

Original entry on oeis.org

1, 0, -1, -1, 1, 0, 0, -1, 1, 0, 1, -2, 1, -1, 2, -2, 2, -3, 3, -3, 4, -4, 5, -6, 6, -6, 8, -9, 9, -11, 12, -13, 14, -16, 19, -19, 21, -25, 26, -28, 32, -36, 38, -41, 46, -50, 55, -60, 65, -70, 77, -85, 91, -99, 108, -116, 126, -138, 149, -160, 174, -188, 202, -219, 237, -255, 274, -296

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000005, A000040, A000586, A000607, A001221, A008683, A083399, A088705, A280954, A305614.

nn=20;
ser=Product[1/(1+x^p),{p,Select[Range[nn],PrimeQ]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}] (* Gus Wiseman, Jun 06 2018 *)

a(n) = A184198(n) - A184199(n). - Vaclav Kotesovec, Jan 11 2021

A055212 Number of composite divisors of n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Jun 23 2000

Trivially, there is only one run of three consecutive 0's. However, there are infinitely many runs of three consecutive 1's and they are at positions A056809(n), A086005(n), and A115393(n) for n >= 1. - Timothy L. Tiffin, Jun 21 2021

			a[20] = 3 because the composite divisors of 20 are 4, 10, 20.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A083399.
Cf. A000005, A001221, A008578 (indices of 0's), A033273, A056809, A086005, A115393, A137944, A137945, A344713.
Cf. A001620, A077761.

a055212 = subtract 1 . a033273  -- Reinhard Zumkeller, Sep 15 2015

Table[ Count[ PrimeQ[ Divisors[n] ], False] - 1, {n, 1, 105} ]
Table[Count[Divisors[n],?CompositeQ],{n,120}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 09 2018 *)
a[n_] := DivisorSigma[0, n] - PrimeNu[n] - 1; Array[a, 100] (* Amiram Eldar, Jun 18 2022 *)

a(n) = numdiv(n) - omega(n) - 1; \\ Michel Marcus, Oct 17 2015

a(n) = A033273(n) - 1.
a(n) = tau(n)-omega(n)-1, where tau=A000005 and omega=A001221. - Reinhard Zumkeller, Jun 13 2003
G.f.: -x/(1 - x) + Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - Ilya Gutkovskiy, Mar 21 2017
Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 2 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - Amiram Eldar, Dec 07 2023

A111725 Number of residues modulo n of the maximum order.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 2, 4, 4, 8, 2, 6, 4, 6, 4, 10, 7, 8, 4, 6, 6, 12, 4, 8, 8, 12, 8, 8, 6, 12, 6, 8, 8, 16, 6, 12, 12, 8, 10, 22, 8, 12, 8, 16, 8, 24, 6, 16, 14, 18, 12, 28, 8, 16, 8, 24, 16, 24, 12, 20, 16, 30, 8, 24, 14, 24, 12, 16, 18, 24, 8, 24, 24, 18, 16, 40, 14, 32, 12

Offset: 1

Max Alekseyev, Nov 18 2005

nonn

The maximum order modulo n is given by A002322(n).
a(n) is the number of primitive lambda-roots of n. - Michel Marcus, Mar 17 2016
A primitive lambda-root is an element of maximal order modulo n. - Joerg Arndt, Mar 19 2016
a(n) is odd if and only if n is a factor of 24, i.e., n is in A018253. - Jianing Song, Apr 27 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots, 2009.
P. J. Cameron and D. A. Preece, Primitive lambda-roots, 2014.
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.
S. R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.
S. Li, On the number of elements with maximal order in the multiplicative group modulo n, Act. Arithm. 86 (2) (1998) 113, Theorem 2.1

Cf. A002322, A008330, A300064, A300065, A300079, A300080.

LiDelta := proc(q,n)
    local a,p,e,lam,v ;
    a := 0 ;
    lam := numtheory[lambda](n) ;
    for p in numtheory[factorset](n) do
        e := padic[ordp](n,p) ;
        if p =2 and e= 3 and q =2 and padic[ordp](lam,q) = 1 then
            return A083399(n) ;
        elif isprime(q) then
            v := padic[ordp](lam,q) ;
            if modp( numtheory[lambda](p^e),q^v) = 0 then
                a := a+1 ;
            end if;
        end if:
    end do:
    a ;
end proc:
A111725 := proc(n)
    local a,q ;
    a := 1;
    for q in numtheory[factorset](numtheory[lambda](n)) do
        a := a*(1-1/q^LiDelta(q,n)) ;
    end do:
    a*numtheory[phi](n) ;
end proc:
seq(A111725(n),n=1..30) ; # R. J. Mathar, Sep 29 2017

f[list_]:=Count[list,Max[list]];Map[f,Table[Table[MultiplicativeOrder[k,n],{k,Select[Range[n],GCD[#,n]==1&]}],{n,1,100}]]  (* Geoffrey Critzer, Jan 26 2013 *)

{ a(n) = my(r, c, r1); r=1; c=0; for(k=0, n-1, if(gcd(k, n)!=1, next); r1=znorder(Mod(k,n)); if(r1==r, c++); if(r1>r, r=r1; c=1) ); c; }

{ A111725(n) = if(n<3,return(1)); my(k,p); k=znstar(n)[2]; p=factor(k[1])[,1]; eulerphi(n) * prod(i=1,#p, (1-1/p[i]^vecsum(apply(x->valuation(k[1]\x,p[i])==0,k))) ); } \\ Max Alekseyev, Oct 23 2021

For prime n, a(n) = phi(phi(n)) = A010554(n) = phi(n-1). - Nick Hobson, Jan 09 2007
Decompose (Z/nZ)* as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then a(n) = Sum_{d divides psi(n)} (mu(psi(n)/d)*Product{i=1..m} gcd(d, k_i)). This is an immediate corollary from the fact that the number of elements in (Z/nZ)* such that x^d == 1 (mod n) is Product{i=1..m} gcd(d, k_i). Here (Z/nZ)* is the multiplicative group of integers modulo n, psi(n) = A002322(n) and mu(n) = A008683(n). - Jianing Song, Apr 27 2019
From Jianing Song, Oct 12 2021: (Start)
Decompose (Z/nZ)* as a product of cyclic groups C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j, then a(n) = phi(n) * Product_{p prime dividing phi(n)} (1 - 1/p^r(p)), where r(p) is the number of j such that v(k_j,p) = v(k_m,p), v(s,p) is the p-adic valuation of s.
Proof: let G = (Z/nZ)*, G_p be the Sylow p-subgroup of G, then G = Product_{p prime dividing phi(n)} G_p: every element x can be uniquely written as Product_{p prime dividing phi(n)} x_p for x_p in G_p. Let ord(x) be the order of x. Since ord(x_p, x_p') = 1 for distinct p and p', we have ord(x) = Product_{p prime dividing phi(n)} ord(x_p), hence x is of maximal order if and only if each x_p is of maximal order in G_p.
Each G_p is isomorphic to C_{p^(e_1)} x C_{p^(e_2)} x ... x C_{p^(e_m)} for e_1 <= e_2 <= ... <= e_m, e_m > 0. Write x_p = (x_{p,1}, x_{p,2}, ..., x_{p,m}). Suppose that e_m = e_{m-1} = ... = e_{m-r+1} > e_{m-r}, then x_p is of maximal order in G_p if and only of x_{p,j} is of order p^(e_m) for some m-r+1 <= j <= m, so the number of such x_p is p^(e_1) * p^(e_2) * ... * p^(e_{m-r}) * (p^(r*e_m) - p^(r*((e_m)-1))) = |G_p| * (1 - 1/p^r).
An example: n = 15903, then (Z/nZ)* = C_6 x C_18 x C_90. We can see that r(2) = 3, r(3) = 2 and r(5) = 1, so a(15903) = phi(15903) * (1 - 1/2^3) * (1 - 1/3^2) * (1 - 1/5^1) = 6048.
It should be clear that a(n) = phi(phi(n)) if and only if r(p) = 1 for every prime p dividing phi(n), or v(k_{m-1},p) < v(k_m,p) for every prime p dividing phi(n). Otherwise, a(n) > phi(phi(n)). (End)

A292586 a(n) = A002110(A001221(n)) = product of first omega(n) primes.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 2, 2, 6, 2, 6, 2, 6, 6, 2, 2, 6, 2, 6, 6, 6, 2, 6, 2, 6, 2, 6, 2, 30, 2, 2, 6, 6, 6, 6, 2, 6, 6, 6, 2, 30, 2, 6, 6, 6, 2, 6, 2, 6, 6, 6, 2, 6, 6, 6, 6, 6, 2, 30, 2, 6, 6, 2, 6, 30, 2, 6, 6, 30, 2, 6, 2, 6, 6, 6, 6, 30, 2, 6, 2, 6, 2, 30, 6, 6, 6, 6, 2, 30, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 2, 30, 2, 6, 30

Offset: 1

Antti Karttunen, Sep 25 2017

nonn

The connection with binary tree A005940 is explained by the fact that on a trajectory from its root (1) to any number n, the numbers of the form 4k+2 will never occur consecutively (they are only born as right children of odd numbers, while all their right descendants from then onward are multiples of four). Thus all the runs are separate runs of length one, from which follows that A278222 when applied to A292382 yields only primorials. Moreover, the steps producing 4k+2 numbers are also only steps in A005940 that add new distinct prime factors to the generated number. Thus the total number of such steps is equal to the number of distinct prime factors of the eventual n. Hence A278222(A292382(n)) = A002110(A001221(n)).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to primorial numbers

Cf. A005940, A086568, A278222, A292382.

Cf. A083399 (restricted growth transform of this sequence).

Array[Product[Prime@ i, {i, PrimeNu@ #}] &, 105] (* Michael De Vlieger, Sep 25 2017 *)

A292586(n) = prod(i=1, omega(n), prime(i));

(define (A292586 n) (A002110 (A001221 n)))
(define (A292586 n) (A278222 (A292382 n)))

a(n) = A002110(A001221(n)).
a(n) = A278222(A292382(n)).
For all n >= 1:
A001221(n) = A001221(a(n)) = A001222(a(n)) = A000120(A292382(n)).

A292582 Restricted growth sequence transform of A292589(n); filter based on the prime signature of {n divided by largest squarefree divisor of n}.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 25 2017

nonn

Also a restricted growth sequence transform of A278222(A292380(n)): a filter constructed from the runlengths of multiples of 4 encountered in trajectories of A005940-tree.

Antti Karttunen, Table of n, a(n) for n = 1..65536
Index entries for sequences computed from exponents in factorization of n

Cf. A003557, A005940, A046523, A083399, A278222, A292380, A292583, A292585, A292586, A292587, A292589.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
write_to_bfile(1,rgs_transform(vector(16384,n,A046523(A003557(n)))),"b292582.txt");

A345926 Number of distinct possible alternating sums of permutations of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3

Offset: 1

Gus Wiseman, Jul 14 2021

nonn

First differs from A096825 at a(90) = 3, A096825(90) = 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.
Also the number of possible values of A056239(d) where d is a divisor of n with half as many prime factors (rounded up) as n.

			Grouping the 12 permutations of {1,2,2,3} by alternating sum k gives:
  k = -2: (1223) (1322) (2213) (2312)
  k =  0: (1232) (2123) (2321) (3212)
  k =  2: (2132) (2231) (3122) (3221)
so a(90) = 3.

The version for prime factors instead of indices is A343943.
A000005 counts divisors.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices,  row sums of A112798.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by length and alternating sum.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A008549, A032443, A083399, A096825, A239830, A344607, A344609, A344651, A345957, A345960, A345961.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Union[ats/@Permutations[primeMS[n]]]],{n,100}]

from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A345926(n):
    fs = dict((primepi(a),b) for (a,b) in factorint(n).items())
    return len(set(sum(d) for d in multiset_combinations(fs, (sum(fs.values())+1)//2))) # Chai Wah Wu, Aug 23 2021

A106404 Number of even semiprimes dividing n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0

Offset: 1

Reinhard Zumkeller, May 02 2005

nonn

Also the number of prime divisors p|n such that n/p is even. - Gus Wiseman, Jun 06 2018

			a(60) = #{4, 6, 10} = #{2*2, 2*3, 2*5} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000607, A001221, A001227, A008683, A083399, A088705, A100484, A205745, A305614.

a106404 n = length [d | d <- takeWhile (<= n) a100484_list, mod n d == 0]
-- Reinhard Zumkeller, Jan 31 2012

Table[Length[Select[Divisors[n],PrimeQ[#]&&EvenQ[n/#]&]],{n,100}] (* Gus Wiseman, Jun 06 2018 *)
Table[Count[Divisors[n],?(EvenQ[#]&&PrimeOmega[#]==2&)],{n,110}] (* _Harvey P. Dale, May 04 2021 *)
a[n_] := If[EvenQ[n], PrimeNu[n/2], 0]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

a(n)=if(n%2,0,omega(n/2)) \\ Charles R Greathouse IV, Jan 30 2012

def A106404(n):
    return add(1-(n/d)%2 for d in divisors(n) if is_prime(d))
print([A106404(n) for n in (1..105)])  # Peter Luschny, Jan 30 2012

a(n) = A086971(n) - A106405(n).
a(A100484(n)) = 1.
a(A005408(n)) = 0.
a(A005843(n)) > 0 for n>1.
a(2n) = omega(n), a(2n+1) = 0, where omega(n) is the number of distinct prime divisors of n, A001221. - Franklin T. Adams-Watters, Jun 09 2006
a(n) = card { d | d*p = n, d even, p prime }. - Peter Luschny, Jan 30 2012
O.g.f.: Sum_{p prime} x^(2p)/(1 - x^(2p)). - Gus Wiseman, Jun 06 2018

cp's OEIS Frontend

A341444 Dirichlet inverse of A083399, where A083399(n) = 1 + omega(n).

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A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.

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A305614 Expansion of Sum_{p prime} x^p/(1 + x^p).

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A048165 Expansion of Product_{k > 0} 1/(1 + x^prime(k)).

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A055212 Number of composite divisors of n.

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A111725 Number of residues modulo n of the maximum order.

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