A001221 -id:A001221 - cp's OEIS frontend

A007774 Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118

Offset: 1

Luke Pebody (ltp1000(AT)hermes.cam.ac.uk)

nonn

Every group of order p^a * q^b is solvable (Burnside, 1904). - Franz Vrabec, Sep 14 2008

Characteristic function for a(n): floor(omega(n)/2) * floor(2/omega(n)) where omega(n) is the number of distinct prime factors of n. - Wesley Ivan Hurt, Jan 10 2013

			20 is a term because 20 = 2^2*5 with two distinct prime divisors 2, 5.

T. D. Noe, Table of n, a(n) for n = 1..1000
W. Burnside, On groups of order p^alpha q^beta, Proc. London Math. Soc. (2) 1 (1904), 388-392.
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.

Subsequence of A085736; A256617 is a subsequence.
Row 2 of A125666.
Cf. A001358 (products of two primes), A014612 (products of three primes), A014613 (products of four primes), A014614 (products of five primes), where the primes are not necessarily distinct.
Cf. A006881, A046386, A046387, A067885 (product of exactly 2, 4, 5, 6 distinct primes respectively).
Cf. A000040, A112801.

a007774 n = a007774_list !! (n-1)
a007774_list = filter ((== 2) . a001221) [1..]
-- Reinhard Zumkeller, Aug 02 2012

with(numtheory,factorset):f := proc(n) if nops(factorset(n))=2 then RETURN(n) fi; end;

Select[Range[0,6! ],Length[FactorInteger[ # ]]==2&] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2010 *)
Select[Range[120],PrimeNu[#]==2&] (* Harvey P. Dale, Jun 03 2020 *)

is(n)=omega(n)==2 \\ Charles R Greathouse IV, Apr 01 2013

from sympy import primefactors
A007774_list = [n for n in range(1,10**5) if len(primefactors(n)) == 2] # Chai Wah Wu, Aug 23 2021

Expanded definition. - N. J. A. Sloane, Aug 22 2021

A013939 Partial sums of sequence A001221 (number of distinct primes dividing n).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 34, 36, 37, 39, 40, 43, 44, 45, 47, 49, 51, 53, 54, 56, 58, 60, 61, 64, 65, 67, 69, 71, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 93, 96, 97, 99, 101, 102, 104, 107, 108, 110, 112

Offset: 1

N. J. A. Sloane, Henri Lifchitz

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A005187, A006218, A011371, A013936.
Cf. A093614, A048865, A006218.
Cf. A027748, A000040, A082287.
Cf. A022559.
Cf. A077761.
Cf. A346617.
Cf. A001222, A048803.

a013939 n = a013939_list !! (n-1)
a013939_list = scanl1 (+) $ map a001221 [1..]
-- Reinhard Zumkeller, Feb 16 2012

[(&+[Floor(n/NthPrime(k)): k in [1..n]]): n in [1..70]]; // G. C. Greubel, Nov 24 2018

A013939 := proc(n) option remember;  `if`(n = 1, 0, a(n) + iquo(n+1, ithprime(n+1))) end:
seq(A013939(i), i = 1..69);  # Peter Luschny, Jul 16 2011

a[n_] := Sum[Floor[n/Prime[k]], {k, 1, n}]; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Jun 11 2012, from 2nd formula *)
Accumulate[PrimeNu[Range[120]]] (* Harvey P. Dale, Jun 05 2015 *)

t=0;vector(99,n,t+=omega(n)) \\ Charles R Greathouse IV, Jan 11 2012

a(n)=my(s);forprime(p=2,n,s+=n\p);s \\ Charles R Greathouse IV, Jan 11 2012

a(n) = sum(k=1, sqrtint(n), k * (primepi(n\k) - primepi(n\(k+1)))) + sum(k=1, n\(sqrtint(n)+1), if(isprime(k), n\k, 0)); \\ Daniel Suteu, Nov 24 2018

from sympy.ntheory import primefactors
print([sum(len(primefactors(k)) for k in range(1,n+1)) for n in range(1, 121)]) # Indranil Ghosh, Mar 19 2017

from sympy import primerange
def A013939(n): return sum(n//p for p in primerange(n+1)) # Chai Wah Wu, Oct 06 2024

[sum(floor(n/nth_prime(k)) for k in (1..n)) for n in (1..70)] # G. C. Greubel, Nov 24 2018

a(n) = Sum_{k <= n} omega(k).
a(n) = Sum_{k = 1..n} floor( n/prime(k) ).
a(n) = a(n-1) + A001221(n).
a(n) = A093614(n) - A048865(n); see also A006218.
A027748(a(A000040(n))+1) = A000040(n), A082287(a(n)+1) = n.
a(n) = Sum_{k=1..n} pi(floor(n/k)). - Vladeta Jovovic, Jun 18 2006
a(n) = n log log n + O(n). - Charles R Greathouse IV, Jan 11 2012
a(n) = n*(log log n + B) + o(n), where B = 0.261497... is the Mertens constant A077761. - Arkadiusz Wesolowski, Oct 18 2013
G.f.: (1/(1 - x))*Sum_{k>=1} x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 02 2017
a(n) = Sum_{k=1..floor(sqrt(n))} k * (pi(floor(n/k)) - pi(floor(n/(k+1)))) + Sum_{p prime <= floor(n/(1+floor(sqrt(n))))} floor(n/p). - Daniel Suteu, Nov 24 2018
a(n) = Sum_{k>=1} k * A346617(n,k). - Alois P. Heinz, Aug 19 2021
a(n) = A001222(A048803(n+1)). - Flávio V. Fernandes, Jan 14 2025

More terms from Henry Bottomley, Jul 03 2001

A212166 Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Matthew Vandermast, May 22 2012

			36 = 2^2*3^2 has 2 positive exponents in its prime factorization. The maximal exponent in its prime factorization is also 2. Therefore, 36 belongs to this sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Includes subsequences A000040, A006939, A138534, A181555, A181825.
See also A212164, A212165, A212167, A212168.
Cf. A001221, A050326, A051903, A188654 (complement), A225230.

import Data.List (elemIndices)
a212166 n = a212166_list !! (n-1)
a212166_list = map (+ 1) $ elemIndices 0 a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] == Length[f]]; Select[Range[424], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) == #e;} \\ Amiram Eldar, Sep 08 2024

A225230(a(n)) = 0; A050326(a(n)) = 1. - Reinhard Zumkeller, May 03 2013

A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 7, 1, 3, 3, 35, 1, 7, 1, 7, 3, 3, 1, 11, 3, 3, 5, 7, 1, 3, 1, 63, 3, 3, 3, 9, 1, 3, 3, 11, 1, 3, 1, 7, 7, 3, 1, 75, 3, 7, 3, 7, 1, 11, 3, 11, 3, 3, 1, 1, 1, 3, 7, 231, 3, 3, 1, 7, 3, 3, 1, 19, 1, 3, 7, 7, 3, 3, 1, 75, 35, 3, 1, 1, 3, 3, 3, 11, 1, 1, 3, 7, 3, 3, 3, 133, 1, 7, 7, 9, 1, 3, 1, 11, 3

Offset: 1

Antti Karttunen, Aug 12 2018

The first negative term is a(210) = -7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001221, A063524, A046644 (denominators).

Cf. also A317831, A317925, A317933, A317845, A317846, A317936, A317938, A317939.

A317937aux(n) = if(1==n,n,(omega(n)-sumdiv(n,d,if((d>1)&&(dA317937aux(d)*A317937aux(n/d),0)))/2);
A317937(n) = numerator(A317937aux(n));

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
DirSqrt(v)={my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dAndrew Howroyd, Aug 13 2018

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A001221(n) - Sum_{d|n, d>1, d 1.

A049060 a(n) = (-1)^omega(n)Sum_{d|n} d(-1)^omega(d), where omega(n) = A001221(n) is number of distinct primes dividing n.

Original entry on oeis.org

1, 1, 2, 5, 4, 2, 6, 13, 11, 4, 10, 10, 12, 6, 8, 29, 16, 11, 18, 20, 12, 10, 22, 26, 29, 12, 38, 30, 28, 8, 30, 61, 20, 16, 24, 55, 36, 18, 24, 52, 40, 12, 42, 50, 44, 22, 46, 58, 55, 29, 32, 60, 52, 38, 40, 78, 36, 28, 58, 40, 60, 30, 66, 125, 48, 20, 66, 80, 44, 24, 70

Offset: 1

N. J. A. Sloane

Might be called (-1)sigma(n). If x = Product p_i^r_i, then (-1)sigma(x) = Product (-1 + Sum p_i^s_i, s_i = 1 to r_i) = Product ((p_i^(r_i+1)-1)/(p_i-1)-2), with (-1)sigma(1) = 1. - Yasutoshi Kohmoto, May 23 2005

R. J. Mathar, Table of n, a(n) for n = 1..100000

Used in A049057, A049058, A049059.

Cf. A000203, A001221, A057723, A060640, A126602, A126690.

A049060 := proc(n) local it, ans, i, j; it := ifactors(n): ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(-1+sum(ifactors(n)[2][i][1]^j, j=1..ifactors(n)[2][i][2])): od: RETURN(ans) end: [seq(A049060(i),i=1..n)];

a[p_?PrimeQ] := p-1; a[1] = 1; a[n_] := Times @@ ((#[[1]]^(#[[2]] + 1) - 2*#[[1]] + 1)/(#[[1]] - 1) & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, May 21 2012 *)

A049060(n)={ local(i,resul,rmax,p) ; if(n==1, return(1) ) ; i=factor(n) ; rmax=matsize(i)[1] ; resul=1 ; for(r=1,rmax, p=0 ; for(j=1,i[r,2], p += i[r,1]^j ; ) ; resul *= p-1 ; ) ; return(resul) ; } { for(n=1,40, print(n," ",A049060(n)) ) ; } \\ R. J. Mathar, Oct 12 2006

apply( A049060(n)=vecprod([(f[1]^(f[2]+1)-1)\(f[1]-1)-2 | f<-factor(n)~]), [1..99]) \\ M. F. Hasler, Sep 21 2022

from math import prod
from sympy import factorint
def A049060(n): return prod((p**(e+1)-2*p+1)//(p-1) for p,e in factorint(n).items()) # Chai Wah Wu, Sep 13 2021

a(n) = Sum_{d|n} d*(-1)^A001221(d).
Multiplicative with a(p^e) = (p^(e+1)-2*p+1)/(p-1).
Simpler: a(p^e) = (p^(e+1)-1)/(p-1)-2. - M. F. Hasler, Sep 21 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 2/p^2 + 2/p^3) = 0.4478559359... . - Amiram Eldar, Oct 25 2022

More terms from James Sellers, May 03 2000

Better description from Vladeta Jovovic, Apr 06 2002

A062799 Inverse Möbius transform of the numbers of distinct prime factors (A001221).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 12, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 20, 1, 4, 7, 6, 4, 12, 1, 7, 4, 12, 1, 17, 1, 4, 7, 7, 4, 12, 1, 13, 4, 4

Offset: 1

Labos Elemer, Jul 19 2001

nonn

Let us say that two divisors d_1 and d_2 of n are adjacent divisors if d_1/d_2 or d_2/d_1 is a prime. Then a(n) is the number of all pairs of adjacent divisors of n. - Vladimir Shevelev, Aug 16 2010
Equivalent to the preceding comment: a(n) is the number of edges in the directed multigraph on tau(n) vertices, vertices labeled by the divisors d_i of n, where edges connect vertex(d_i) and vertex(d_j) if the ratio of the labels is a prime. - R. J. Mathar, Sep 23 2011
a(A001248(n)) = 2. - Reinhard Zumkeller, Dec 02 2014
Depends on the prime signature of n as follows: a(A025487(n)) = 0, 1, 2, 4, 3, 7, 4, 10, 12, 5, 12, 13, 20, 6, 17, 16, 28, 7, 22, 33, 19 ,32, 24, 36, 8, 27, 46, ... (n>=1). - R. J. Mathar, May 28 2017

			n = 255: divisors = {1, 3, 5, 15, 17, 51, 85, 255}, a(255) = 0+1+1+2+1+2+2+3 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.
E. Pérez Herrero, Psychedelic Geometry Blogspot, CURIOUS SERIES-002

Cf. A001221, A001248, A027750.

a062799 = sum . map a001221 . a027750_row
-- Reinhard Zumkeller, Dec 02 2014

read("transforms") ;
A001221 := proc(n)
        nops(numtheory[factorset](n)) ;
end proc:
omega := [seq(A001221(n),n=1..80)] ;
ones := [seq(1,n=1..80)] ;
DIRICHLET(ones,omega) ; # R. J. Mathar, Sep 23 2011
N:= 1000: # to get a(1) to a(N)
B:= Vector(N,t-> nops(numtheory:-factorset(t))):
A:= Vector(N):
for d from 1 to N do
  md:= d*[$1..floor(N/d)];
  A[md]:= map(`+`,A[md],B[d])
od:
convert(A,list); # Robert Israel, Oct 21 2015

f[n_] := Block[{d = Divisors[n], c = l = 0, k = 2}, l = Length[d]; While[k < l + 1, c = c + Length[ FactorInteger[ d[[k]] ]]; k++ ]; Return[c]]; Table[f[n], {n, 1, 100} ]
omega[n_] := Length[FactorInteger[n]]; SetAttributes[omega, Listable]; omega[1] := 0; A062799[n_] := Plus @@ omega[Divisors[n]] (* Enrique Pérez Herrero, Sep 08 2009 *)

a(n)=my(f=factor(n)[,2],s);forvec(v=vector(#f,i,[0,f[i]]),s+=sum(i=1,#f,v[i]>0));s \\ Charles R Greathouse IV, Oct 15 2015

vector(100, n, sumdiv(n, k, omega(k))) \\ Altug Alkan, Oct 15 2015

a(n) = Sum_{d|n} A001221(d), that is, where d runs over divisors of n.
For squarefree s (i.e., s in A005117), a(s) = omega(s)*2^(omega(s)-1), where omega(n) = A001221(n). Also, for n>1, a(n) <= omega(n)*A000005(n) - 1. - Enrique Pérez Herrero, Sep 08 2009
Let n=Product_{i=1..omega(n)} p(i)^e(i). a(n) = d[Product_{i=1..omega(n)} (1 + e(i)*x)]/dx|x=1. In other words, a(n) = Sum_{m>=1} A146289(n,m)*m. - Geoffrey Critzer, Feb 10 2015
a(A000040(n)) = 1; a(A001248(n)) = 2; a(A030078(n)) = 3; a(A030514(n)) = 4; a(A050997(n)) = 5. - Altug Alkan, Oct 17 2015
a(n) = Sum_{prime p|n} A000005(n/p). - Max Alekseyev, Aug 11 2016
G.f.: Sum_{k>=1} omega(k)*x^k/(1 - x^k), where omega(k) is the number of distinct primes dividing k (A001221). - Ilya Gutkovskiy, Jan 16 2017
Dirichlet g.f.: zeta(s)^2*primezeta(s) where primezeta(s) = Sum_{prime p} p^(-s). - Benedict W. J. Irwin, Jul 16 2018

A141809 Irregular table: Row n (of A001221(n) terms, for n>=2) consists of the largest powers that divides n of each distinct prime that divides n. Terms are arranged by the size of the distinct primes. Row 1 = (1).

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 3, 7, 8, 9, 2, 5, 11, 4, 3, 13, 2, 7, 3, 5, 16, 17, 2, 9, 19, 4, 5, 3, 7, 2, 11, 23, 8, 3, 25, 2, 13, 27, 4, 7, 29, 2, 3, 5, 31, 32, 3, 11, 2, 17, 5, 7, 4, 9, 37, 2, 19, 3, 13, 8, 5, 41, 2, 3, 7, 43, 4, 11, 9, 5, 2, 23, 47, 16, 3, 49, 2, 25, 3, 17, 4, 13, 53, 2, 27, 5, 11, 8, 7, 3

Offset: 1

Leroy Quet, Jul 07 2008

In other words, except for row 1, row n contains the unitary prime power divisors of n, sorted by the prime. - Franklin T. Adams-Watters, May 05 2011

A034684(n) = smallest term of n-th row; A028233(n) = T(n,1); A053585(n) = T(n,A001221(n)); A008475(n) = sum of n-th row for n > 1. - Reinhard Zumkeller, Jan 29 2013

			60 has the prime factorization 2^2 * 3^1 * 5^1, so row 60 is (4,3,5).
From _M. F. Hasler_, Oct 12 2018: (Start)
The table starts:
    n : largest prime powers dividing n
    1 :  1
    2 :  2
    3 :  3
    4 :  4
    5 :  5
    6 :  2, 3
    7 :  7
    8 :  8
    9 :  9
   10 :  2, 5
   11 : 11
   12 :  4, 3
   etc. (End)

Reinhard Zumkeller, Rows n=1..10000 of triangle, flattened
Eric Weisstein's World of Mathematics, Prime Factorization

A027748, A124010 are used in a formula defining this sequence.
Cf. A001221 (row lengths), A008475 (row sums), A028233 (column 1), A034684 (row minima), A053585 (right edge).
Cf. A060175, A141810, A213925.

a141809 n k = a141809_row n !! (k-1)
a141809_row 1 = [1]
a141809_row n = zipWith (^) (a027748_row n) (a124010_row n)
a141809_tabf = map a141809_row [1..]
-- Reinhard Zumkeller, Mar 18 2012

f[{x_, y_}] := x^y; Table[Map[f, FactorInteger[n]], {n, 1, 50}] // Grid (* Geoffrey Critzer, Apr 03 2015 *)

A141809_row(n)=if(n>1, [f[1]^f[2]|f<-factor(n)~], [1]) \\ M. F. Hasler, Oct 12 2018, updated Aug 19 2022

T(n,k) = A027748(n,k)^A124010(n,k) for n > 1, k = 1..A001221(n). - Reinhard Zumkeller, Mar 15 2012

A278908 Multiplicative with a(p^e) = 2^omega(e), where omega = A001221.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 30 2016

The number of exponential unitary (or e-unitary) divisors of n and the number of exponential squarefree exponential divisors (or e-squarefree e-divisors) of n. These are divisors of n = Product p(i)^a(i) of the form Product p(i)^b(i) where each b(i) is a unitary divisor of a(i) in the first case, or each b(i) is a squarefree divisor of a(i) in the second case. - Amiram Eldar, Dec 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zahi, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.7, eq (20).
Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.
László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.
Xiangzhen Zhao, Min Liu, and Yu Huang, Mean value for the function t^(e)(n) over square-full numbers, Scientia Magna, Vol. 8, No. 3 (2012), pp. 110-114.
Index entries for sequences computed from exponents in factorization of n

Cf. A001221.

A278908 := proc(n)
    local a,p,e;
    a := 1;
    if n =1 then
        ;
    else
        for p in ifactors(n)[2] do
            e := op(2,p) ;
            a := a*2^A001221(e) ;
        end do:
    end if;
    a ;
end proc:

Table[Times @@ Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> 2^PrimeNu[e]], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2^omega(f[k,2]); f[k,2] = 1); factorback(f); \\ Michel Marcus, Jul 28 2017

(define (A278908 n) (if (= 1 n) n (* (A000079 (A001221 (A067029 n))) (A278908 (A028234 n))))) ;; Antti Karttunen, Jul 27 2017

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Product_{p prime} (1 + Sum_{k>=2} (2*omega(k) - 2^omega(k-1))/p^k) = 1.5431653193... (Tóth, 2007). - Amiram Eldar, Nov 08 2020

A113901 Product of omega(n) and bigomega(n) = A001221(n)*A001222(n), where omega(x): number of distinct prime divisors of x. bigomega(x): number of prime divisors of x, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 6, 1, 4, 4, 4, 1, 6, 1, 6, 4, 4, 1, 8, 2, 4, 3, 6, 1, 9, 1, 5, 4, 4, 4, 8, 1, 4, 4, 8, 1, 9, 1, 6, 6, 4, 1, 10, 2, 6, 4, 6, 1, 8, 4, 8, 4, 4, 1, 12, 1, 4, 6, 6, 4, 9, 1, 6, 4, 9, 1, 10, 1, 4, 6, 6, 4, 9, 1, 10, 4, 4, 1, 12, 4, 4, 4, 8, 1, 12, 4, 6, 4, 4, 4, 12, 1, 6, 6, 8, 1, 9

Offset: 1

Cino Hilliard, Jan 29 2006

Positions of first appearances are A328964. - Gus Wiseman, Nov 05 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000

A307409(n) is (bigomega(n) - 1) * omega(n).
A328958(n) is sigma_0(n) - bigomega(n) * omega(n).
Cf. A000005, A001221, A001222, A060687, A066921, A066922, A068993, A070175, A071625, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965.

Table[PrimeNu[n]*PrimeOmega[n], {n,1,50}] (* G. C. Greubel, Apr 23 2017 *)

PARI
```
a(n) = omega(n)*bigomega(n);
```

a(n) = 1 iff n is prime.
A068993(a(n)) = 4. - Reinhard Zumkeller, Mar 13 2011
a(n) = A066921(n)*A066922(n). - Amiram Eldar, May 07 2025

A092248 Parity of number of distinct primes dividing n (function omega(n)) parity of A001221.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1

Offset: 1

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 19 2004

a(p^r) = 1 for all primes p and all exponents r>0. - Tom Edgar, Mar 22 2015

			For n = 1, 0 primes divide 1 so a(1)=0.
For n = 2, there is 1 distinct prime dividing 2 (itself) so a(2)=1.
For n = 4 = 2^2, there is 1 distinct prime dividing 4 so a(4)=1.
For n = 5, there is 1 distinct prime dividing 5 (itself) so a(5)=1.
For n = 6 = 2*3, there are 2 distinct primes dividing 6 so a(6)=0.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for characteristic functions

Cf. A001221, A268411.

Table[Boole[OddQ[PrimeNu[n]]], {n, 1, 100}] (* Geoffrey Critzer, Feb 16 2015 *)

for (i=1,200,if(Mod(omega(i),2)==0,print1(0,","),print1(1,",")))

from sympy import primefactors
def a(n): return 0 if n==1 else 1*(len(primefactors(n))%2==1) # Indranil Ghosh, Jun 01 2017

If omega(n) is even then a(n) = 0 else a(n) = 1. By convention, a(1) = 0. (Also because A001221(1) = 0 is an even number too).
a(n) = A000035(A001221(n)). - Michel Marcus, Mar 22 2015
a(n) = A268411(A156552(n)). - Antti Karttunen, May 30 2017

Offset corrected by Reinhard Zumkeller, Oct 03 2008

cp's OEIS Frontend

A007774 Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Extensions

A013939 Partial sums of sequence A001221 (number of distinct primes dividing n).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Sage

Formula

Extensions

A212166 Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A317937 Numerators of sequence whose Dirichlet convolution with itself yields sequence A001221 (omega n) + A063524 (1, 0, 0, 0, ...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A049060 a(n) = (-1)^omega(n)*Sum_{d|n} d*(-1)^omega(d), where omega(n) = A001221(n) is number of distinct primes dividing n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A062799 Inverse Möbius transform of the numbers of distinct prime factors (A001221).

Views

Author

Keywords

Comments

Examples

A049060 a(n) = (-1)^omega(n)Sum_{d|n} d(-1)^omega(d), where omega(n) = A001221(n) is number of distinct primes dividing n.