A050997 -id:A050997 - cp's OEIS frontend

A325249 Sum of the omega-sequence of n.

0, 1, 1, 3, 1, 5, 1, 4, 3, 5, 1, 8, 1, 5, 5, 5, 1, 8, 1, 8, 5, 5, 1, 9, 3, 5, 4, 8, 1, 7, 1, 6, 5, 5, 5, 7, 1, 5, 5, 9, 1, 7, 1, 8, 8, 5, 1, 10, 3, 8, 5, 8, 1, 9, 5, 9, 5, 5, 1, 12, 1, 5, 8, 7, 5, 7, 1, 8, 5, 7, 1, 10, 1, 5, 8, 8, 5, 7, 1, 10, 5, 5, 1, 12, 5

Offset: 1

Gus Wiseman, Apr 16 2019

nonn

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1).

			The omega-sequence of 180 is (5,3,2,2,1) with sum 13, so a(180) = 13.

Positions of m's are A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7).
Cf. A056239, A118914, A181819, A182850, A323023, A325274.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number).

omseq[n_Integer]:=If[n<=1,{},Total/@NestWhileList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Total[#]>1&]];
Table[Total[omseq[n]],{n,100}]

a(n) = A056239(A325248(n)).

a(n!) = A325274(n).

A179665 a(n) = prime(n)^9.

Original entry on oeis.org

512, 19683, 1953125, 40353607, 2357947691, 10604499373, 118587876497, 322687697779, 1801152661463, 14507145975869, 26439622160671, 129961739795077, 327381934393961, 502592611936843, 1119130473102767

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

Product_{n >= 2, m_n = (a(n) mod 4) - 2} ((a(n) + 1) / (a(n) - 1))^m_n = 209865342976 / 209844223875. - Dimitris Valianatos, May 13 2020

			a(1) = 512 since the ninth power of the first prime is 2^9 = 512. - _Wesley Ivan Hurt_, Mar 27 2014

T. D. Noe, Table of n, a(n) for n = 1..1000
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Will Nicholes, Prime Signatures.

Cf. A001248, A030078, A030514, A050997, A030516, A092759, A179645.

Cf. A013667, A013676, A085969.

[p^9: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A179665:=n->ithprime(n)^9; seq(A179665(n), n=1..30); # Wesley Ivan Hurt, Mar 27 2014

Array[Prime[ # ]^9&, 30]
Prime[Range[30]]^9 (* Harvey P. Dale, Jul 20 2024 *)

a(n)=prime(n)^9 \\ Charles R Greathouse IV, Jul 20 2011

a(n) = A000040(n)^9 = A001017(A000040(n)). - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = P(9) = 0.0020044675... (A085969). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(9)/zeta(18) = A013667/A013676.
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(9) = 1/A013667. (End)

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

A080257 Numbers having at least two distinct or a total of at least three prime factors.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Feb 10 2003

Complement of A000430; A080256(a(n)) > 3.
A084114(a(n)) > 0, see also A084110.
Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
Numbers with at least 4 divisors. - Franklin T. Adams-Watters, Jul 28 2006
Union of A024619 and A033942; A211110(a(n)) > 2. - Reinhard Zumkeller, Apr 02 2012
Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - Gus Wiseman, Jul 03 2019
Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - Joshua Zelinsky, Jan 23 2025

			8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
From _Gus Wiseman_, Jul 03 2019: (Start)
The sequence of terms together with their prime indices begins:
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
(End)

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

Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.

Cf. A060687, A118914, A323014, A323023, A325249.

a080257 n = a080257_list !! (n-1)
a080257_list = m a024619_list a033942_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
                           | x == y = x : m xs ys
                           | x > y  = y : m xs' ys
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* Harvey P. Dale, Jul 23 2013 *)

PARI
```
is(n)=omega(n)>1 || isprimepower(n)>2
```

is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = n + O(n/log n). - Charles R Greathouse IV, Sep 14 2015

Definition clarified by Harvey P. Dale, Jul 23 2013

A335988 Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.

Original entry on oeis.org

1, 8, 27, 32, 125, 128, 216, 243, 343, 512, 864, 1000, 1331, 1944, 2048, 2187, 2197, 2744, 3125, 3375, 3456, 4000, 4913, 6859, 7776, 8192, 9261, 10648, 10976, 12167, 13824, 16000, 16807, 17496, 17576, 19683, 24389, 25000, 27000, 29791, 30375, 31104, 32768, 35937

Offset: 1

Amiram Eldar, Jul 03 2020

nonn

This sequence is a permutation of A355038.
This sequence is also a permutation of the exponentially odd numbers (A268335) multiplied by the square of their squarefree kernel (A007947).
a(n)/rad(a(n)) is a permutation of the squares.
a(n)/rad(a(n))^2 is a permutation of the exponentially odd numbers.

			8 = 2^3 is a term since the exponent of its prime factor 2 is 3 which is odd and larger than 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A001694 and A268335.
Intersection of A036966 and A268335.
A355038 in ascending order.
A030078, A050997, A092759, A179665, A079395 and A138031 are subsequences.
Cf. A007947, A065487.

Join[{1}, Select[Range[10^5], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]

from math import isqrt, prod
from sympy import factorint
def afind(N): # all terms up to limit N
    cands = (n**2*prod(factorint(n**2)) for n in range(1, isqrt(N//2)+2))
    return sorted(c for c in cands if c <= N)
print(afind(4*10**4)) # Michael S. Branicky, Jun 16 2022

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911... (A065487).

A056595 Number of nonsquare divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 6, 1, 3, 2, 4, 1, 7, 1, 3, 3, 3, 3, 5, 1, 3, 3, 6, 1, 7, 1, 4, 4, 3, 1, 7, 1, 4, 3, 4, 1, 6, 3, 6, 3, 3, 1, 10, 1, 3, 4, 3, 3, 7, 1, 4, 3, 7, 1, 8, 1, 3, 4, 4, 3, 7, 1, 7, 2, 3, 1, 10, 3, 3, 3, 6, 1, 10, 3, 4, 3, 3, 3, 9, 1, 4, 4, 5, 1, 7, 1

Offset: 1

Labos Elemer, Jul 21 2000

nonn

a(A000430(n))=1; a(A030078(n))=2; a(A030514(n))=2; a(A006881(n))=3; a(A050997(n))=3; a(A030516(n))=3; a(A054753(n))=4; a(A000290(n))=A055205(n). - Reinhard Zumkeller, Aug 15 2011

			a(36)=5 because the set of divisors of 36 has tau(36)=nine elements, {1, 2, 3, 4, 6, 9, 12, 18, 36}, five of which, that is {2, 3, 6, 12, 18}, are not perfect squares.

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

Cf. A000005, A000188, A046951.
See A194095 and A194096 for record values and where they occur.
Cf. A001620, A013661.

a056595 n = length [d | d <- [1..n], mod n d == 0, a010052 d == 0]
-- Reinhard Zumkeller, Aug 15 2011

A056595 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if not issqr(d) then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A056595(n),n=1..40) ; # R. J. Mathar, Aug 18 2024

Table[Count[Divisors[n],?(#!=Floor[Sqrt[#]]^2&)],{n,110}] (* _Harvey P. Dale, Jul 10 2013 *)
a[1] = 0; a[n_] := Times @@ (1 + (e = Last /@ FactorInteger[n])) - Times @@ (1 + Floor[e/2]); Array[a, 100] (* Amiram Eldar, Jul 22 2019 *)

a(n)=sumdiv(n,d,!issquare(d)) \\ Charles R Greathouse IV, Aug 28 2016

a(n) = A000005(n) - A046951(n) = tau(n) - tau(A000188(n)).

Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - zeta(2) - 1)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A030515 Numbers with exactly 6 divisors.

Original entry on oeis.org

12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428

Offset: 1

Jeff Burch

Numbers which are either the 5th power of a prime or the product of a prime and the square of a different prime, i.e., numbers which are in A050997 (5th powers of primes) or A054753. - Henry Bottomley, Apr 25 2000
Also numbers which are the square root of the product of their proper divisors. - Amarnath Murthy, Apr 21 2001
Such numbers are multiplicatively 3-perfect (i.e., the product of divisors of a(n) equals a(n)^3). - Lekraj Beedassy, Jul 13 2005
Since A119479(6)=5, there are never more than 5 consecutive terms. Quintuples of consecutive terms start at 10093613546512321, 14414905793929921, 266667848769941521, ... (A141621). No such quintuple contains a term of the form p^5. - Ivan Neretin, Feb 08 2016

Amarnath Murthy, A note on the Smarandache Divisor sequences, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

R. J. Mathar, Table of n, a(n) for n = 1..1000
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, 1.12.
Eric Weisstein's World of Mathematics, Divisor Product

Cf. A061117.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, {2,seq(i,i=3..floor(N/4))}):
S:= select(`<=`,{seq(p^5, p = Primes),seq(seq(p*q^2, p=Primes minus {q}),q=Primes)},N):
sort(convert(S,list)); # Robert Israel, Feb 10 2016

f[n_]:=Length[Divisors[n]]==6; lst={};Do[If[f[n],AppendTo[lst,n]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
Select[Range[500],DivisorSigma[0,#]==6&] (* Harvey P. Dale, Oct 02 2014 *)

is(n)=numdiv(n)==6 \\ Charles R Greathouse IV, Jan 23 2014

from sympy import divisor_count
def ok(n): return divisor_count(n) == 6
print([k for k in range(429) if ok(k)]) # Michael S. Branicky, Dec 18 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A030515(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p**2) for p in primerange(isqrt(x)+1))+primepi(integer_nthroot(x,3)[0])-primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Union of A050997 and A054753. - Lekraj Beedassy, Jul 13 2005

A000005(a(n))=6. - Juri-Stepan Gerasimov, Oct 10 2009

Definition clarified by Jonathan Sondow, Jan 23 2014

A133527 Sum of fifth powers of four consecutive primes.

Original entry on oeis.org

20207, 181226, 552276, 1969008, 4428300, 10703592, 30843448, 58052742, 124920600, 234340458, 360837752, 561553608, 910405144, 1509473242, 2207061100, 3327841200, 4713875058, 6072022352, 8304482450, 10893397986

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=20207 because 2^5+3^5+5^5+7^5=20207.

Cf. A034963, A133524, A133525, A133526, A133528.

a = 5; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a, {n, 1, 100}]

a(n) = A133532(n) + A050997(n+3). - Michel Marcus, Nov 08 2013

A133532 Sum of fifth powers of three consecutive primes.

Original entry on oeis.org

3400, 20175, 180983, 549151, 1952201, 4267249, 10332299, 29423591, 55576643, 118484257, 213829309, 332208601, 492209651, 794548943, 1362464799, 1977716093, 2909645707, 3998950759, 5227426051, 6954357343, 9089168635

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=3400 because 2^5+3^5+5^5=3400.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133533.

a = 5; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a, {n, 1, 100}]
Total[#^5]&/@Partition[Prime[Range[30]],3,1] (* Harvey P. Dale, May 26 2011 *)

a(n) = A133536(n) + A050997(n+2). - Michel Marcus, Nov 09 2013

A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 147, 148, 149

Offset: 1

Chris Boyd, Sep 14 2013

nonn

No term is the product of two other terms.
Squares of terms and pairwise products of distinct terms form a subsequence of A028260.
Numbers n such that A162642(n) = 1. - Jason Kimberley, Oct 10 2016
Numbers k such that A007913(k) is a prime number. - Amiram Eldar, Jul 27 2020

Chris Boyd, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes, IX. On the set of integers representable as a product of a prime and square, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.

Subsequence of A026424.
Cf. A028260, A162642, A229153.
Cf. A007913, A013661.
Subsequences: A000040, A030078, A050997, A054753, A092759, A179643, A179665, A246551.

With[{nn=70},Take[Union[Flatten[Table[p*m^2,{p,Prime[Range[nn]]},{m,nn}]]], nn]] (* Harvey P. Dale, Dec 02 2014 *)

test(n)=isprime(core(n))
for(n=1,200,if(test(n), print1(n",")))

from math import isqrt
from sympy import primepi
def A229125(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//y**2) for y in range(1,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962). - Amiram Eldar, Jul 27 2020

cp's OEIS Frontend

A325249 Sum of the omega-sequence of n.

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A179665 a(n) = prime(n)^9.

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

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A080257 Numbers having at least two distinct or a total of at least three prime factors.

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A335988 Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.

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A056595 Number of nonsquare divisors of n.

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