A083342 -id:A083342 - cp's OEIS frontend

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

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

A136141 Decimal expansion of Sum_{p prime} 1/(p*(p-1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 09 2008

Excess of prime factors with multiplicity over distinct prime factors for random (large) integers. - Charles R Greathouse IV, Sep 06 2011
Sum of reciprocals of (proper) prime powers. The sum of reciprocals of all proper powers is A072102. - Charles R Greathouse IV, Apr 24 2012
Decimal expansion of the infinite sum of the reciprocals of the prime powers which are not prime (A246547). - Robert G. Wilson v, May 13 2019
See the second 'Applications' example under the Mathematica help file for the function PrimePowerQ. - Robert G. Wilson v, May 13 2019
It easy to prove that this constant < 1 (Sum_{p prime} 1/(p*(p-1)) < Sum_{k>=2} 1/(k*(k-1)) = 1). Luthar (1969) asks for a better upper bound. The solution shows that this constant is < 3/2 - log(2) = 0.80685... . - Amiram Eldar, Feb 14 2025

			Equals 1/2 + 1/(3*2) + 1/(5*4) + 1/(7*6) + ...
= 0.7731566690497951278643674598559423956187413360831860483110060673567...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants, p. 94.

Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (first 1002 terms from Jason Kimberley).
Henri Cohen, High-precision computation of Hardy-Littlewood constants, preprint 1991.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Stephan Ramon Garcia and Ethan Simpson Lee, Explicit conditional bounds for the residue of a Dedekind zeta-function at s = 1, arXiv:2506.17416 [math.NT], 2025. See p. 3.
R. S. Luthar, Problem E 2192, Elementary Problems, The American Mathematical Monthly, Vol. 76, No. 8 (1969), p. 938; An Upper Bound, Solution to Problem E 2192, by O. P. Lossers, ibid., Vol. 77, No. 7 (1970), pp. 769-770.
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Tables 8 and 10.
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
Index to constants which are prime zeta sums {1,1,0}

Cf. A152447 (over the semiprimes), A000040, A000720, A001248, A046660 (excess, see first comment), A072102, A077761, A083342, A179119, A246547.

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).

R := RealField(105);
c := &+[R|(EulerPhi(n)-MoebiusMu(n))/n*Log(ZetaFunction(R,n)):n in[2..360]];
Reverse(IntegerToSequence(Floor(c*10^103))); // Jason Kimberley, Jan 12 2017

digits = 103; sp = NSum[PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 2*digits]; RealDigits[sp, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)

W(x)=solve(y=log(x)/2,max(1,log(x)),y*exp(y)-x)
eps()=2. >> (32*ceil(default(realprecision)/9.63))
primezeta(s)=my(t=s*log(2),iter=W(t/eps())\t);sum(k=1,iter, moebius(k)/k*log(abs(zeta(k*s))))
a(lim,e)={ \\ choose parameters to maximize speed and precision
    my(x,y=exp(W(lim)-.5));
    x=lim^e*(e*log(y))^e*(y*log(y))^-e*incgam(-e,e*log(y));
    forprime(p=2,lim,x+=1/((p*1.)^e*(p-1)));
    x+sum(n=2,e,primezeta(n))
}; \\ Charles R Greathouse IV, Sep 07 2011

sumeulerrat(1/(p*(p-1))) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{n>=1} 1/A036689(n).
Equals Sum_{s>=2} P(s), where P is the prime zeta function. - Charles R Greathouse IV, Sep 06 2011
Equals A083342 - A077761, that is, Sum_{n>=2} ((EulerPhi(n) - MoebiusMu(n))/n) * log(zeta(n)). - Jean-François Alcover, Sep 02 2015
Equals 2 * Sum_{k>=2} pi(k)/(k^3-k), where pi(k) = A000720(k) (Shamos, 2011, p. 8). - Amiram Eldar, Mar 12 2024

More terms from D. S. McNeil, Sep 06 2011

More digits from Jean-François Alcover, Sep 02 2015

A087436 Number of odd prime factors of n, counted with repetitions.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 0, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 3, 0, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 2

Offset: 1

Reinhard Zumkeller, Sep 03 2003

Number of parts larger than 1 in the partition with Heinz number n. The Heinz number of an integer partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: a(9) = 2 because the partition with Heinz number 9 (=3*3) is [2,2]. - Emeric Deutsch, Oct 02 2015

Totally additive because both A001222 and A007814 are. a(2) = 0, and a(p) = 1 for odd primes p, a(m*n) = a(m)+a(n) for m, n > 1. - Antti Karttunen, Jul 10 2020

			a(9) = 2 because 9 = 3*3 has 2 odd prime factors. - _Emeric Deutsch_, Oct 02 2015

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from G. C. Greubel)

Cf. A000244 (the first occurrence of each n, and also the positions of records).

Cf. A000265, A001222, A005087, A007814, A083342, A215366, A336118, A336161.

seq(bigomega(n) - padic[ordp](n, 2), n=1..102); # Peter Luschny, Dec 06 2017

Join[{0},Table[Length[Select[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]],OddQ]],{n,2,110}]] (* Harvey P. Dale, Feb 01 2013 *)

a(n) = bigomega(n) - valuation(n, 2); \\ Michel Marcus, Sep 10 2019

A087436(n) = (bigomega(n>>valuation(n,2))); \\ Antti Karttunen, Jul 10 2020

a(n) = A001222(n) - A007814(n).
a(n) = A001222(A000265(n)). - Antti Karttunen, Jul 10 2020
Sum_{k=1..n} a(k) = n * (log(log(n)) + B_2 - 1) + O(n/log(n)), where B_2 = A083342. - Amiram Eldar, May 16 2025

A303279 Expansion of (1/(1 - x)^2) * Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).

Original entry on oeis.org

0, 1, 3, 7, 12, 19, 27, 38, 51, 66, 82, 101, 121, 143, 167, 195, 224, 256, 289, 325, 363, 403, 444, 489, 536, 585, 637, 692, 748, 807, 867, 932, 999, 1068, 1139, 1214, 1290, 1368, 1448, 1532, 1617, 1705, 1794, 1886, 1981, 2078, 2176, 2279, 2384, 2492, 2602, 2715, 2829, 2947, 3067

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

Sum of exponents in prime-power factorization of product of first n factorials (A000178).

Partial sums of A022559.

			a(5) = 12 because 2!*3!*4!*5! = 2^8*3^3*5 and 8 + 3 + 1 = 12.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathématique, Vol. 115, No. 129 (2024), pp. 45-76.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Eric Weisstein's World of Mathematics, Superfactorial.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to factorial numbers.

Cf. A000178, A001222, A022559, A083342, A303281.

Half of row sums of triangle A356718.

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[bigomega](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Oct 07 2021

nmax = 55; Rest[CoefficientList[Series[1/(1 - x)^2 Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[PrimeOmega[BarnesG[n + 2]], {n, 55}]
Table[ Sum[ PrimeOmega@ j, {k, n}, {j, k}], {n, 55}]

a(n) = my(t=0); sum(k=1, n, t+=bigomega(k)); \\ Daniel Suteu, Jan 17 2019

a(n) = (n^2/2) * (log(log(n)) + c + O(1/log(n))), where c = A083342 (De Koninck and Verreault, 2024, p. 49, eq. (3.1)). - Amiram Eldar, Dec 10 2024

a(n) = (1/2) * Sum_{k=0..n} A356718(n,k). - Dario T. de Castro, Feb 19 2025

A350387 a(n) is the sum of the odd exponents in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 28 2021

First differs from A125073 at n = 32.

a(n) is the number of prime divisors of n, counted with multiplicity, with an odd exponent in the prime factorization of n.

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

Cf. A000290, A001222, A001620, A083342, A125073, A162641, A268335, A350386, A350389.

f[p_, e_] := If[OddQ[e], e, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = my(f=factor(n)); sum(k=1, #f~, if (f[k,2] %2, f[k,2])); \\ Michel Marcus, Dec 28 2021

from sympy import factorint
def a(n): return sum(e for e in factorint(n).values() if e%2 == 1)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Dec 28 2021

Additive with a(p^e) = e if e is odd and 0 otherwise.
a(n) = A001222(A350389(n)).
a(n) = 0 if and only if n is a positive square (A000290 \ {0}).
a(n) = A001222(n) - A350386(n).
a(n) = A001222(n) if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = A083342 - Sum_{p prime} 2*p/((p-1)*(p+1)^2) = gamma + Sum_{p prime} (log(1-1/p) + (p^2+1)/((p-1)*(p+1)^2)) = 0.2384832800... and gamma is Euler's constant (A001620).

A361205 a(n) = 2*omega(n) - bigomega(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, -1, 0, 2, 1, 1, 1, 2, 2, -2, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, -1, 1, 1, 3, 1, -3, 2, 2, 2, 0, 1, 2, 2, 0, 1, 3, 1, 1, 1, 2, 1, -1, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 2, 1, 2, 1, -4, 2, 3, 1, 1, 2, 3, 1, -1, 1, 2, 1, 1, 2, 3, 1, -1, -2, 2, 1, 2

Offset: 1

Gus Wiseman, Mar 16 2023

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

Without doubling omega we have -A046660.
Positions of 0's are A067801, counted by A239959.
Positions of negative terms are A360558, counted by A360254.
Positions of nonpositive terms are A361204, counted by A237363.
Positions of positive terms are A361393, counted by A237365.
Positions of nonnegative terms are A361395, counted by A361394.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
Cf. A061395, A067340, A111907, A324517, A324522, A326567/A326568.
Cf. A077761, A083342, A136141.

Table[2*PrimeNu[n]-PrimeOmega[n],{n,100}]

Additive with a(p^e) = 2 - e. - Amiram Eldar, Mar 26 2023

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = 2*A077761 - A083342 = A077761 - A136141 = -0.511659... . - Amiram Eldar, Oct 01 2023

A080256 Sum of numbers of distinct and of all prime factors of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 10 2003

a(n) = 2 iff n is prime, A000040; a(n) > 2 iff n is composite, A002808; a(n) <= 3 iff n is prime or square of prime, A000430; a(n) = 3 iff n is square of prime, A001248; a(A080257(n)) > 3;

a(n) <= 4 iff product of proper divisors <= n^2, A007964; a(n) = 4 iff n has four divisors, A030513; a(n) > 4 iff product of proper divisors > n^2, A058080; a(A064598(n)) <= 5; a(A080258(n)) = 5.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000040, A000430, A002808, A001248, A007964, A058080, A064598, A080257, A080258.

Cf. A001221, A001222, A046660, A077761, A083342.

f[n_] := Plus @@ (Last /@ FactorInteger[n] + 1); Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 03 2005 *)

a(n) = {my(f = factor(n)); omega(f) + bigomega(f);} \\ Amiram Eldar, Sep 28 2023

a(n) = Omega(n) + omega(n) = A001221(n) + A001222(n).
Additive with a(p^e) = e + 1.
Sum_{k=1..n} a(k) = 2 * n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 + A083342 = 1.29615109474508069537... . - Amiram Eldar, Sep 28 2023

A076399 Number of prime factors of n-th perfect power (with repetition).

Original entry on oeis.org

0, 2, 3, 2, 4, 2, 3, 5, 4, 2, 6, 4, 4, 2, 3, 7, 6, 2, 4, 6, 4, 5, 8, 2, 6, 3, 2, 6, 4, 4, 9, 2, 8, 4, 4, 6, 6, 2, 6, 2, 6, 10, 4, 4, 4, 8, 3, 2, 4, 4, 8, 2, 9, 6, 2, 6, 6, 11, 4, 7, 3, 2, 10, 4, 6, 4, 6, 6, 2, 8, 4, 5, 8, 4, 4, 6, 2, 8, 2, 4, 6, 12, 4, 6, 2, 6, 4, 6, 3, 2, 10, 2, 4, 6, 6, 9, 4, 6, 2, 10, 8

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A001222, A025478, A025479, A076398, A076400, A083342.

a076399 n = a001222 (a025478 n) * a025479 n
-- Reinhard Zumkeller, Mar 28 2014

PrimeOmega[Select[Range[10^4], # == 1 || GCD @@ FactorInteger[#][[;; , 2]] > 1 &]] (* Amiram Eldar, Feb 18 2023 *)

is(n) = n==1 || ispower(n);
apply(bigomega, select(is, [1..5000])) \\ Amiram Eldar, Feb 18 2023

from sympy import mobius, integer_nthroot, primeomega
def A076399(n):
    def f(x): return int(n-2+x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return int(primeomega(kmax)) # Chai Wah Wu, Aug 14 2024

a(n) = A001222(A001597(n)).
a(n) = A001222(A025478(n))*A025479(n).
Sum_{A001597(k) <= x} a(k) = 2*sqrt(x)*log(log(x)) + 2*(B_2 - log(2))*sqrt(x) + O(sqrt(x)/log(x)), where B_2 = A083342 (Jakimczuk and Lalín, 2022). - Amiram Eldar, Feb 18 2023, corrected Sep 21 2024

A210241 Partial sums of A073093.

Original entry on oeis.org

1, 3, 5, 8, 10, 13, 15, 19, 22, 25, 27, 31, 33, 36, 39, 44, 46, 50, 52, 56, 59, 62, 64, 69, 72, 75, 79, 83, 85, 89, 91, 97, 100, 103, 106, 111, 113, 116, 119, 124, 126, 130, 132, 136, 140, 143, 145, 151, 154, 158, 161, 165, 167, 172, 175, 180, 183, 186, 188

Offset: 1

Reinhard Zumkeller, Mar 19 2012

nonn

Number of terms in the first n rows of triangle A210208.

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

Cf. A022559, A073093, A083342, A210208.

a210241 n = a210241_list !! (n-1)
a210241_list = scanl1 (+) a073093_list

Accumulate[Array[PrimeOmega[#] + 1 &, 100]] (* Amiram Eldar, Apr 05 2025 *)

a(n) = A022559(n) + n.

a(n) = n * (log(log(n)) + B_2 + 1) + O(n/log(n)), where B_2 = A083342. - Amiram Eldar, Apr 05 2025

A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 15 2021

The total number of prime powers (not including 1) that divide n is A001222(n).

The least number k such that a(k) = m is A122756(m).

			12 has 4 bi-unitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.

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

Cf. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.
Similar sequences: A001222, A349258, A349281.
Cf. A083342, A179119.

f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ Amiram Eldar, Sep 29 2023

Additive with a(p^e) = e if e is odd, and e-1 if e is even.
a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
a(n) = A001222(n) - A162641(n). - Amiram Eldar, May 18 2023
From Amiram Eldar, Sep 29 2023: (Start)
a(n) = A001222(A350390(n)) (the number of prime factors of the largest exponentially odd number dividing n, counted with multiplicity).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)

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