A144494 -id:A144494 - cp's OEIS frontend

A001222 Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 3, 1, 4, 3, 2, 1, 5, 1, 3, 2

Offset: 1

N. J. A. Sloane

Maximal number of terms in any factorization of n.
Number of prime powers (not including 1) that divide n.
Sum of exponents in prime-power factorization of n. - Daniel Forgues, Mar 29 2009
Sum_{d|n} 2^(-A001221(d) - a(n/d)) = Sum_{d|n} 2^(-a(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Row sums in A067255. - Reinhard Zumkeller, Jun 11 2013
Conjecture: Let f(n) = (x+y)^a(n), and g(n) = x^a(n), and h(n) = (x+y)^A046660(n) * y^A001221(n) with x, y complex numbers and 0^0 = 1. Then f(n) = Sum_{d|n} g(d)*h(n/d). This is proved for x = 1-y (see Dressler and van de Lune link). - Werner Schulte, Feb 10 2018
Let r, s be some fixed integers. Then we have:
  (1) The sequence b(n) = Dirichlet convolution of r^bigomega(n) and s^bigomega(n) is multiplicative with b(p^e) = (r^(e+1)-s^(e+1))/(r-s) for prime p and e >= 0. The case r = s leads to b(p^e) = (e+1)*r^e.
  (2) The sequence c(n) = Dirichlet convolution of r^bigomega(n) and mu(n)*s^bigomega(n) is multiplicative with c(p^e) = (r-s)*r^(e-1) and c(1) = 1 for prime p and e > 0 where mu(n) = A008683(n). - Werner Schulte, Feb 20 2019
a(n) is also the length of the composition series for every solvable group of order n. - Miles Englezou, Apr 25 2024

			16=2^4, so a(16)=4; 18=2*3^2, so a(18)=3.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 119, #12, omega(n).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)
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], p. 844.
Benoit Cloitre, A tauberian approach to RH, arXiv:1107.0812 [math.NT], 2011.
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406.
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
Douglas E. Iannucci and Urban Larsson, Game values of arithmetic functions, arXiv:2101.07608 [math.NT], 2021. Section 1.1.1. pp. 4-5.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.10.
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Roundness
Wolfram Research, First 50 numbers factored
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences computed from indices in prime factorization

Cf. A001221 (omega, primes counted without multiplicity), A008836 (Liouville's lambda, equal to (-1)^a(n)), A046660, A144494, A074946, A134334.
Bisections give A091304 and A073093. A086436 is essentially the same sequence. Cf. A022559 (partial sums), A066829 (parity), A092248 (parity of omega).
Sequences listing n such that a(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (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. A079149 (primes adj. to integers with at most 2 prime factors, a(n)<=2).
Cf. A000120, A020639, A091202, A091222, A156552, A267116, A268387.
Cf. A027748 (without repetition).
Cf. A000010.

Concatenation([0],List([2..150],n->Length(Factors(n)))); # Muniru A Asiru, Feb 21 2019

import Math.NumberTheory.Primes.Factorisation (factorise)
a001222 = sum . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n, distinct=false) for n in 1:60])  # Peter Luschny, Jan 02 2024

[n eq 1 select 0 else &+[p[2]: p in Factorization(n)]: n in [1..120]]; // Bruno Berselli, Nov 27 2013

with(numtheory): seq(bigomega(n), n=1..111);

Array[ Plus @@ Last /@ FactorInteger[ # ] &, 105]
PrimeOmega[Range[120]] (* Harvey P. Dale, Apr 25 2011 *)

PARI
```
vector(100,n,bigomega(n))
```

from sympy import primeomega
def a(n): return primeomega(n)
print([a(n) for n in range(1, 112)]) # Michael S. Branicky, Apr 30 2022

[sloane.A001222(n) for n in (1..120)] # Giuseppe Coppoletta, Jan 19 2015

[gp.bigomega(n) for n in range(1,131)] # G. C. Greubel, Jul 13 2024

(define (A001222 n) (let loop ((n n) (z 0)) (if (= 1 n) z (loop (/ n (A020639 n)) (+ 1 z)))))
;; Requires also A020639 for which an equally naive implementation can be found under that entry. - Antti Karttunen, Apr 12 2017

n = Product_(p_j^k_j) -> a(n) = Sum_(k_j).
Dirichlet g.f.: ppzeta(s)*zeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s-1) and ppzeta(s) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
Totally additive with a(p) = 1.
a(n) = if n=1 then 0 else a(n/A020639(n)) + 1. - Reinhard Zumkeller, Feb 25 2008
a(n) = Sum_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(n) = A022559(n) - A022559(n-1).
G.f.: Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Jan 25 2017
a(n) = A091222(A091202(n)) = A000120(A156552(n)). - Antti Karttunen, circa 2004 and Mar 06 2017
a(n) >= A267116(n) >= A268387(n). - Antti Karttunen, Apr 12 2017
Sum_{k=1..n} 2^(-A001221(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001221(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = a(A046523(n)) = A007814(A108951(n)) = A061395(A122111(n)) = A056239(A181819(n)) = A048675(A293442(n)). - Antti Karttunen, Apr 30 2022

More terms from David W. Wilson

A001221 Number of distinct primes dividing n (also called omega(n)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

From Peter C. Heinig (algorithms(AT)gmx.de), Mar 08 2008: (Start)
This is also the number of maximal ideals of the ring (Z/nZ,+,*). Since every finite integral domain must be a field, every prime ideal of Z/nZ is a maximal ideal and since in general each maximal ideal is prime, there are just as many prime ideals as maximal ones in Z/nZ, so the sequence gives the number of prime ideals of Z/nZ as well.
The reason why this number is given by the sequence is that the ideals of Z/nZ are precisely the subgroups of (Z/nZ,+). Hence for an ideal to be maximal it has form a maximal subgroup of (Z/nZ,+) and this is equivalent to having prime index in (Z/nZ) and this is equivalent to being generated by a single prime divisor of n.
Finally, all the groups arising in this way have different orders, hence are different, so the number of maximal ideals equals the number of distinct primes dividing n. (End)
Equals double inverse Mobius transform of A143519, where A051731 = the inverse Mobius transform. - Gary W. Adamson, Aug 22 2008
a(n) is the number of unitary prime power divisors of n (not including 1). - Jaroslav Krizek, May 04 2009 [corrected by Ilya Gutkovskiy, Oct 09 2019]
Sum_{d|n} 2^(-A001221(d) - A001222(n/d)) = Sum_{d|n} 2^(-A001222(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Up to 2*3*5*7*11*13*17*19*23*29  - 1 = 6469693230 - 1, also the decimal expansion of the constant 0.01111211... = Sum_{k>=0} 1/(10 ^ A000040(k) - 1) (see A073668). - Eric Desbiaux, Jan 20 2014
The average order of a(n): Sum_{k=1..n} a(k) ~ Sum_{k=1..n} log log k. - Daniel Forgues, Aug 13-16 2015
From Peter Luschny, Jul 13 2023: (Start)
We can use A001221 and A001222 to classify the positive integers as follows.
   A001222(n) = A001221(n) = 0 singles out {1}.
Restricting to n > 1:
   A001222(n)^A001221(n) = 1: A000040, prime numbers.
   A001221(n)^A001222(n) = 1: A246655, prime powers.
   A001222(n)^A001221(n) > 1: A002808, the composite numbers.
   A001221(n)^A001222(n) > 1: A024619, complement of A246655.
   n^(A001222(n) - A001221(n)) = 1: A144338, products of distinct primes. (End)
Inverse Möbius transform of the characteristic function of primes (A010051). - Wesley Ivan Hurt, Jun 22 2024
Dirichlet convolution of A010051(n) and 1. - Wesley Ivan Hurt, Jul 15 2025

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.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/Cambridge University Press London 1935.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 from NJAS)
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].
Henry Bottomley, Prime factors calculator
J. Brennen, Prime Factoring Applet
J. Britton, Prime Factorization Machine
A. Dendane, Prime Factors Calculator
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406
J. Flament, Decomposition d'un nombre en facteurs premiers
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
A. Hodges, Java Applet for Factorisation
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
S. O. S. Math, Prime factorization of the first 1000 integers
K. Matthews, Factorization and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
J. Moyer, "Prime Factors of Integers" server for numbers up to 10^36
Primefan, The First 2500 Integers,Factored
Primefan, Factorer
F. Richman, Factoring into Primes
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Wikipedia, Prime factor
Wikipedia, Table of prime factors
G. Xiao, WIMS server, Factoris
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Cf. A001222 (primes counted with multiplicity), A046660, A285577, A346617. Partial sums give A013939.
Cf. also A125666, A069010, A087624, A091202, A091221, A143519, A144494, A158210, A156542, A156552, A000010, A008683.
Sum of the k-th powers of the primes dividing n for k=0..10: this sequence (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k=0..10: this sequence (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

import Math.NumberTheory.Primes.Factorisation (factorise)
a001221 = length . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n) for n in 1:60]) # Peter Luschny, Jan 02 2024

[#PrimeDivisors(n): n in [1..120]]; // Bruno Berselli, Oct 15 2021

A001221 := proc(n) local t1, i; if n = 1 then return 0 else t1 := 0; for i to n do if n mod ithprime(i) = 0 then t1 := t1 + 1 end if end do end if; t1 end proc;
A001221 := proc(n) nops(numtheory[factorset](n)) end proc: # Emeric Deutsch
omega := n -> NumberTheory:-NumberOfPrimeFactors(n, 'distinct'): # Peter Luschny, Jun 15 2025

Array[ Length[ FactorInteger[ # ] ]&, 100 ]
PrimeNu[Range[120]]  (* Harvey P. Dale, Apr 26 2011 *)

func(nops(numlib::primedivisors(n)), n):

numlib::omega(n)$ n=1..110 // Zerinvary Lajos, May 13 2008

PARI
```
a(n)=omega(n)
```

from sympy.ntheory import primefactors
print([len(primefactors(n)) for n in range(1, 1001)])  # Indranil Ghosh, Mar 19 2017

def A001221(n): return sum(1 for p in divisors(n) if is_prime(p))
[A001221(n) for n in (1..80)] # Peter Luschny, Feb 01 2012

[sloane.A001221(n) for n in (1..111)] # Giuseppe Coppoletta, Jan 19 2015

[gp.omega(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024

G.f.: Sum_{k>=1} x^prime(k)/(1-x^prime(k)). - Benoit Cloitre, Apr 21 2003; corrected by Franklin T. Adams-Watters, Sep 01 2009
Dirichlet generating function: zeta(s)*primezeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Additive with a(p^e) = 1.
a(1) = 0, a(p) = 1, a(pq) = 2, a(pq...z) = k, a(p^k) = 1, where p, q, ..., z are k distinct primes and k natural numbers. - Jaroslav Krizek, May 04 2009
a(n) = log_2(Sum_{d|n} mu(d)^2). - Enrique Pérez Herrero, Jul 09 2012
a(A002110(n)) = n, i.e., a(prime(n)#) = n. - Jean-Marc Rebert, Jul 23 2015
a(n) = A091221(A091202(n)) = A069010(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
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
a(n) = log_2(Sum_{k=1..n} mu(gcd(n,k))^2/phi(n/gcd(n,k))) = log_2(Sum_{k=1..n} mu(n/gcd(n,k))^2/phi(n/gcd(n,k))), where phi = A000010 and mu = A008683. - Richard L. Ollerton, May 13 2021
Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001222(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-A001222(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = A005089(n) + A005091(n) + A059841(n) = A005088(n) +A005090(n) +A079978(n). - R. J. Mathar, Jul 22 2021
From Wesley Ivan Hurt, Jun 22 2024: (Start)
a(n) = Sum_{p|n, p prime} 1.
a(n) = Sum_{d|n} c(d), where c = A010051. (End)

A022559 Sum of exponents in prime-power factorization of n!.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 8, 11, 13, 15, 16, 19, 20, 22, 24, 28, 29, 32, 33, 36, 38, 40, 41, 45, 47, 49, 52, 55, 56, 59, 60, 65, 67, 69, 71, 75, 76, 78, 80, 84, 85, 88, 89, 92, 95, 97, 98, 103, 105, 108, 110, 113, 114, 118, 120, 124, 126, 128, 129, 133, 134, 136, 139

Offset: 0

Karen E. Wandel (kw29(AT)evansville.edu)

Partial sums of Omega(n) (A001222). - N. J. A. Sloane, Feb 06 2022

			For n=5, 5! = 120 = 2^3*3^1*5^1 so a(5) = 3+1+1 = 5. - _N. J. A. Sloane_, May 26 2018

Daniel Forgues, Table of n, a(n) for n = 0..100000
Mehdi Hassani, On the decomposition of n! into primes, arXiv:math/0606316 [math.NT], 2006-2007.
Keith Matthews, Computing the prime-power factorization of n!
Daniel Suteu, Perl program

Cf. A001222, A013939, A046660, A144494, A115627, A238002, A069513.

a022559 n = a022559_list !! n
a022559_list = scanl (+) 0 $ map a001222 [1..]
-- Reinhard Zumkeller, Feb 16 2012

with(numtheory):with(combinat):a:=proc(n) if n=0 then 0 else bigomega(numbperm(n)) fi end: seq(a(n), n=0..63); # Zerinvary Lajos, Apr 11 2008
# Alternative:
ListTools:-PartialSums(map(numtheory:-bigomega, [$0..200])); # Robert Israel, Dec 21 2018

Array[Plus@@Last/@FactorInteger[ #! ] &, 5!, 0] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)
f[n_] := If[n <= 1, 0, Total[FactorInteger[n]][[2]]]; Accumulate[Array[f, 100, 0]] (* T. D. Noe, Apr 11 2011 *)
Table[PrimeOmega[n!], {n, 0, 70}] (* Jean-François Alcover, Jun 08 2013 *)
Join[{0}, Accumulate[PrimeOmega[Range[70]]]] (* Harvey P. Dale, Jul 23 2013 *)

PARI
```
a(n)=bigomega(n!)
```

first(n)={my(k=0); vector(n, i, k+=bigomega(i))}

a(n) = sum(k=1, primepi(n), (n - sumdigits(n, prime(k))) / (prime(k)-1)); \\ Daniel Suteu, Apr 18 2018

a(n) = my(res = 0); forprime(p = 2, n, cn = n; while(cn > 0, res += (cn \= p))); res \\ David A. Corneth, Apr 27 2018

from sympy import factorint as pf
def aupton(nn):
    alst = [0]
    for n in range(1, nn+1): alst.append(alst[-1] + sum(pf(n).values()))
    return alst
print(aupton(63)) # Michael S. Branicky, Aug 01 2021

a(n) = a(n-1) + A001222(n).
A027746(a(A000040(n))+1) = A000040(n). A082288(a(n)+1) = n.
A001221(n!) = omega(n!) = pi(n) = A000720(n).
a(n) = Sum_{i = 1..n} A001222(i). - Jonathan Vos Post, Feb 10 2010
a(n) = n log log n + B_2 * n + o(n), with B_2 = A083342. - Charles R Greathouse IV, Jan 11 2012
a(n) = A210241(n) - n for n > 0. - Reinhard Zumkeller, Mar 23 2012
G.f.: (1/(1 - x))*Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Mar 15 2017
a(n) = Sum_{k=1..floor(sqrt(n))} k * (A025528(floor(n/k)) - A025528(floor(n/(k+1)))) + Sum_{k=1..floor(n/(floor(sqrt(n))+1))} floor(n/k) * A069513(k). - Daniel Suteu, Dec 21 2018
a(n) = Sum_{prime p<=n} Sum_{k=1..floor(log_p(n))} floor(n/p^k). - Ridouane Oudra, Nov 04 2022
a(n) = Sum_{k=1..n} A069513(k)*floor(n/k). - Ridouane Oudra, Oct 04 2024

Typo corrected by Daniel Forgues, Nov 16 2009

A174024 List of primes of the form x^2+y^2 such that tau(x^2+y^2) = bigomega(x*y).

Original entry on oeis.org

13, 17, 29, 37, 53, 101, 173, 197, 293, 677, 1373, 2213, 4493, 5333, 5477, 8837, 9413, 10613, 17957, 18773, 21317, 26573, 27893, 37253, 42437, 54293, 76733, 85853, 94253, 97973, 98597, 100493, 106277, 120413, 139133, 148997, 214373, 217157

Offset: 1

Michel Lagneau, Mar 05 2010

nonn

bigomega(n) is the number of prime divisors of n (counted with multiplicity) (A001222) Because n = x^2+y^2 is prime, tau(n)= 2, and if we suppose x < y, then (x,y) = (2, p) with p prime or (x,y)=(1, 2q) with q prime.

			13 = 2^2 + 3^2, bigomega(2*3) = 2.
17 = 1+4^2, bigomega(1*4) = 2.
994013 = 2^2 + 997^2, bigomega(2*997) = 2.

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.
J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/Cambridge University Press London 1935.

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].
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A020882, A002313, A001222, A001221 (primes counted without multiplicity), A046660, A144494.

with(numtheory):T:=array(0..50000000):U=array(0..50000000 ): k:=1:for x from 1 to 1000 do:for y from x to 1000 do:if tau(x^2+y^2)= bigomega(x*y) and type(x^2+y^2,prime)=true then T[k]:=x^2+y^2:k:=k+1:else fi:od :od:mini:=T[1]:ii:=1: for p from 1 to k-1 do:for n from 1 to k-1 do:if T[n]< mini then mini:= T[n]:ii:=n: indice:=U[n]: else fi:od:print(mini):T[ii]:= 99999999: ii:=1:mini:=T[1] :od:

A175204 Smallest index m such that omega(m) + omega(m+1) + omega(m+2) = n.

Original entry on oeis.org

1, 2, 4, 10, 20, 68, 154, 644, 2210, 6578, 35308, 92378, 310154, 1042404, 5617820, 35515634, 184055430, 1082950218, 5386096364, 19304763268, 254772473240, 1383442606194

Offset: 2

Michel Lagneau, Mar 04 2010

The arithmetic function omega(m) + omega(m+1) + omega(m+2) = Sum_{j=0..2} A001221(m+j) starts 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 4, 5, 5, 5, 4, 4, 4, 5, 5 (m >= 1).
The sequence is a "first-serve" inverse of this function.
a(24) <= 6127197154440. [Donovan Johnson, Oct 22 2010]

			For n=2, m=1 and omega(1) + omega(2) + omega(3) = 0 + 1 + 1 = 2.
For n=3, m=2 and omega(2) + omega(3) + omega(4) = 1 + 1 + 1 = 3.
For n=4, m=4 and omega(4) + omega(5) + omega(6) = 1 + 1 + 2 = 4.
For n=5, m=10 and omega(10) + omega(11) + omega(12) = 2 + 1 + 2 = 5.
For n=6, m=20 and omega(20) + omega(21) + omega(22) = 2 + 2 + 2 = 6.
For n=7, m=68 and omega(68) + omega(69) + omega(70) = 2 + 2 + 3 = 7.

J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/ Cambridge University Press London 1935.

S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92.
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A001221, A144494.

with(numtheory): for k from 1 to 20 do :indic:=0: for n from 1 to 2000 do :
s1:= ifactors(n)[2] :u1 :=s1[i][1], i=1..nops(s1):uu1:= nops(s1): s2:= ifactors(n+1)[2] :u2 :=s2[i][1], i=1..nops(s2): uu2:= nops(s2): s3:= ifactors(n+2)[2] :u3 :=s3[i][1], i=1..nops(s3): uu3:= nops(s3): if uu1+uu2+uu3 = k and indic=0 then print(n): indic:=1:else fi:od:od:

Added punctuation to the examples. Corrected and edited by Michel Lagneau, Apr 25 2010
Use of variables adapted to OEIS standards by R. J. Mathar, Oct 12 2010
a(16) corrected and a(19)-a(23) from Donovan Johnson, Oct 22 2010

A175206 Smallest number k such that omega(k)+ omega(k+1)+ omega(k+2)+ omega(k+3)= n.

Original entry on oeis.org

1, 2, 3, 9, 12, 33, 75, 153, 492, 987, 4179, 13803, 18444, 134043, 282489, 1013724, 4289592, 12582633, 57495513, 260628717, 801621093, 3307216989, 5313193818, 62909521245, 308935340153, 611063815284

Offset: 3

Michel Lagneau, Mar 04 2010

nonn

omega(.) = A001221(.) is the number of distinct prime divisors of the argument.

a(29) <= 3169630569180. - Donovan Johnson, Sep 17 2011

			For n=3, omega(1)+ omega(2)+ omega(3)+ omega(4)= 0 + 1 + 1 + 1 = 3.
For n=4, omega(2)+ omega(3)+ omega(4)+ omega(5)= 1 + 1 + 1 + 1 = 4.
For n=5, omega(3)+ omega(4)+ omega(5)+ omega(6)= 1 + 1 + 1 + 2 = 5.
For n=6, omega(9)+ omega(10)+ omega(11)+ omega(12)= 1 + 2 + 1 + 2 = 6.
For n=7, omega(12)+ omega(13)+ omega(14)+ omega(15)= 2 + 1 + 2 + 2 = 7.
For n=8, omega(33)+ omega(34)+ omega(35)+ omega(36)= 2 + 2 + 2 + 2 = 8.
For n=9, omega(75)+ omega(76)+ omega(77)+ omega(78)= 2 + 2 + 2 + 3 = 9.

J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/ Cambridge University Press London 1935.

S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92.

Cf. A001222, A001221, A046660, A144494.

with(numtheory):for k from 1 to 20 do : indic:=0: for n from 1 to 1000000000 do :
u1 := nops(factorset(n)): u2 := nops(factorset(n+1)): u3 := nops(factorset(n+2)): u4:= nops(factorset(n+3)):
if u1+u2+u3+u4 = k and indic=0 then print(n): indic:=1: else fi: od: od:

n1=3; nmax=23; omega[k_] := Length[FactorInteger[k]]; Do[om[j] = omega[j], {j, 2, n1+1}]; a[n1]=1; a[n_ /; n>n1] :=  a[n] = (k = a[n-1](* assuming sequence is increasing *); While[k++; Do[om[j] = om[j+1], {j, 1, n1}]; om[n1+1] = omega[k+n1]; Sum[om[j], {j, 1, n1+1}] < n]; k); Table[Print[a[n]]; a[n], {n, n1, nmax}] (* Jean-François Alcover,  Sep 12 2011 *)

Unspecific references removed, offset corrected, variable names standardized - R. J. Mathar, Mar 16 2010

a(24)-a(28) from Donovan Johnson, Sep 17 2011

cp's OEIS Frontend

A055636 Partial sums of A144494.

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A001222 Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).

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A001221 Number of distinct primes dividing n (also called omega(n)).

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A022559 Sum of exponents in prime-power factorization of n!.

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A174024 List of primes of the form x^2+y^2 such that tau(x^2+y^2) = bigomega(x*y).

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