A253905 -id:A253905 - cp's OEIS frontend

A023900 Dirichlet inverse of Euler totient function (A000010).

1, -1, -2, -1, -4, 2, -6, -1, -2, 4, -10, 2, -12, 6, 8, -1, -16, 2, -18, 4, 12, 10, -22, 2, -4, 12, -2, 6, -28, -8, -30, -1, 20, 16, 24, 2, -36, 18, 24, 4, -40, -12, -42, 10, 8, 22, -46, 2, -6, 4, 32, 12, -52, 2, 40, 6, 36, 28, -58, -8, -60, 30, 12, -1, 48, -20, -66, 16, 44, -24, -70, 2, -72, 36, 8, 18, 60, -24, -78, 4, -2

Offset: 1

Olivier Gérard

Also called reciprocity balance of n.
Apart from different signs, same as Sum_{d divides n} core(d)*mu(n/d), where core(d) (A007913) is the squarefree part of d. - Benoit Cloitre, Apr 06 2002
Main diagonal of A191898. - Mats Granvik, Jun 19 2011

			x - x^2 - 2*x^3 - x^4 - 4*x^5 + 2*x^6 - 6*x^7 - x^8 - 2*x^9 + 4*x^10 - ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 125.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
K. Dohmen and M. Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv preprint arXiv:1404.5480 [math.CO], 2014.
R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.

Cf. A000010, A007913, A023898, A117209, A134842.
Moebius transform is A055615.
Cf. A027748, A173557 (gives the absolute values), A295876.
Cf. A253905 (Dgf at s=3).

a023900 1 = 1
a023900 n = product $ map (1 -) $ a027748_row n
-- Reinhard Zumkeller, Jun 01 2015

A023900 := n -> mul(1-i,i=numtheory[factorset](n)); # Peter Luschny, Oct 26 2010

a[ n_] := If[ n < 1, 0, Sum[ d MoebiusMu @ d, { d, Divisors[n]}]] (* Michael Somos, Jul 18 2011 *)
Array[ Function[ n, 1/Plus @@ Map[ #*MoebiusMu[ # ]/EulerPhi[ # ]&, Divisors[ n ] ] ], 90 ]
nmax = 81; Drop[ CoefficientList[ Series[ Sum[ MoebiusMu[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* Stuart Clary, Apr 15 2006 *)
t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] :=  t[n, k] = If[n < k, If[n > 1 && k > 1, Sum[-t[k - i, n], {i, 1, n - 1}], 0], If[n > 1 && k > 1, Sum[-t[n - i, k], {i, 1, k - 1}], 0]]; Table[t[n, n], {n, 36}] (* Mats Granvik, Robert G. Wilson v, Jun 25 2011 *)
Table[DivisorSum[m, # MoebiusMu[#] &], {m, 90}] (* Jan Mangaldan, Mar 15 2013 *)
f[p_, e_] := (1 - p); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

{a(n) = direuler( p=2, n, (1 - p*X) / (1 - X))[n]}

{a(n) = if( n<1, 0, sumdiv( n, d, d * moebius(d)))} /* Michael Somos, Jul 18 2011 */

a(n)=sumdivmult(n,d, d*moebius(d)) \\ Charles R Greathouse IV, Sep 09 2014

from sympy import divisors, mobius
def a(n): return sum([d*mobius(d) for d in divisors(n)]) # Indranil Ghosh, Apr 29 2017

from math import prod
from sympy import primefactors
def A023900(n): return prod(1-p for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

;; With memoization-macro definec.
(definec (A023900 n) (if (= 1 n) 1 (* (- 1 (A020639 n)) (A023900 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = Sum_{ d divides n } d*mu(d) = Product_{p|n} (1-p).
a(n) = 1 / (Sum_{ d divides n } mu(d)*d/phi(d)).
Dirichlet g.f.: zeta(s)/zeta(s-1). - Michael Somos, Jun 04 2000
a(n+1) = det(n+1)/det(n) where det(n) is the determinant of the n X n matrix M_(i, j) = i/gcd(i, j) = lcm(i, j)/j. - Benoit Cloitre, Aug 19 2003
a(n) = phi(n)*moebius(A007947(n))*A007947(n)/n. Logarithmic g.f.: Sum_{n >= 1} a(n)*x^n/n = log(F(x)) where F(x) is the g.f. of A117209 and satisfies: 1/(1-x) = Product_{n >= 1} F(x^n). - Paul D. Hanna, Mar 03 2006
G.f.: A(x) = Sum_{k >= 1} mu(k) k x^k/(1 - x^k) where mu(k) is the Moebius (Mobius) function, A008683. - Stuart Clary, Apr 15 2006
G.f.: A(x) is x times the logarithmic derivative of A117209(x). - Stuart Clary, Apr 15 2006
Row sums of triangle A134842. - Gary W. Adamson, Nov 12 2007
G.f.: x/(1-x) = Sum_{n >= 1} a(n)*x^n/(1-x^n)^2. - Paul D. Hanna, Aug 16 2008
a(n) = phi(rad(n)) *(-1)^omega(n) = A000010(A007947(n)) *(-1)^A001221(n). - Enrique Pérez Herrero, Aug 24 2010
a(n) = Product_{i = 2..n} (1-i)^( (pi(i)-pi(i-1)) * floor( (cos(n*Pi/i))^2 ) ), where pi = A000720, Pi = A000796. - Wesley Ivan Hurt, May 24 2013
a(n) = -limit of zeta(s)*(Sum_{d divides n} moebius(d)/exp(d)^(s-1)) as s->1 for n>1. - Mats Granvik, Jul 31 2013
a(n) = Sum_{d divides n} mu(d)*rad(d), where rad is A007947. - Enrique Pérez Herrero, May 29 2014
Conjecture for n>1: Let n = 2^(A007814(n))*m = 2^(ruler(n))*odd_part(n), where m = A000265(n), then a(n) = (-1)^(m=n)*(0+Sum_{i=1..m and gcd(i,m)=1} (4*min(i,m-i)-m)) = (-1)^(m1} (4*min(i,m-i)-m)). - I. V. Serov, May 02 2017
a(n) = (-1)^A001221(n) * A173557(n). - R. J. Mathar, Nov 02 2017
a(1) = 1; for n > 1, a(n) = (1-A020639(n)) * a(A028234(n)), because multiplicative with a(p^e) = (1-p). - Antti Karttunen, Nov 28 2017
a(n) = 1 - Sum_{d|n, d > 1} d*a(n/d). - Ilya Gutkovskiy, Apr 26 2019
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k)) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)) = 0. (End)
a(n) = rad(n)*(-1)^omega(n)*phi(n)/n = A062953(n)*A000010(n)/n. - Amrit Awasthi, Jan 30 2022
a(n) = mu(n)*phi(n) = A008683(n)*A000010(n) whenever n is squarefree. - Amrit Awasthi, Feb 03 2022
From Peter Bala, Jan 24 2024: (Start)
a(n) = Sum_{d divides n} core(d)*mu(d). Cf. Comment by Benoit Cloitre, dated Apr 06 2002.
a(n) = Sum_{d|n, e|n}  n/gcd(d, e) * mu(n/d) * mu(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value 1 - p for n = p^e, a prime power). (End)
From Peter Bala, Feb 01 2024: (Start)
G.f. Sum_{n >= 1} (2*n-1)*moebius(2*n-1)*x^(2*n-1)/(1 + x^(2n-1)).
a(n) = (-1)^(n+1) * Sum_{d divides n, d odd} d*moebius(d). (End)

A306633 Decimal expansion of zeta(2)/zeta(3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 02 2019

Equals the asymptotic mean of the unitary abundancy index, lim_{n->oo} (1/n) * Sum{k=1..n} usigma(k)/k, where usigma(k) is the sum of the unitary divisors of k (A034448).
From Amiram Eldar, May 12 2023: (Start)
Equals the asymptotic mean of the abundancy index of the squarefree numbers (A005117).
In general, the asymptotic mean of the abundancy index of the k-free numbers (numbers that are not divisible by a k-th power other than 1) is zeta(2)/zeta(k+1) (Jakimczuk and Lalín, 2022). (End)

			1.3684327776202058757367658539847919411308139146524...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link.

Cf. A000010, A001615, A002117, A005117, A013661 (asymptotic mean of sigma(k)/k), A034448, A065463, A253905, A322887.

RealDigits[Zeta[2]/Zeta[3],10, 100][[1]]

zeta(2)/zeta(3) \\ Michel Marcus, Mar 04 2019

Equals A013661/A002117 = 1/A253905.
Equals Sum_{k>=1} phi(k)/k^3, where phi is the Euler totient function (A000010). - Amiram Eldar, Jun 23 2020
Equals Product_{p prime} (1 + 1/(p*(p+1))). - Amiram Eldar, Aug 10 2020
Equals Sum_{k>=1} mu(k)^2/(k*psi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Dedekind psi function values, A001615). - Amiram Eldar, Aug 18 2020

A060800 a(n) = p^2 + p + 1 where p runs through the primes.

Original entry on oeis.org

7, 13, 31, 57, 133, 183, 307, 381, 553, 871, 993, 1407, 1723, 1893, 2257, 2863, 3541, 3783, 4557, 5113, 5403, 6321, 6973, 8011, 9507, 10303, 10713, 11557, 11991, 12883, 16257, 17293, 18907, 19461, 22351, 22953, 24807, 26733, 28057, 30103, 32221

Offset: 1

Jason Earls, Apr 27 2001

Terms are divisible by 3 iff p is of the form 6*m+1 (A002476). - Michel Marcus, Jan 15 2017

			a(3) = 31 because 5^2 + 5 + 1 = 31.

Harry J. Smith, Table of n, a(n) for n = 1..1000
R. J. Mathar, No common terms in the sequences sigma(p^i) and sigma(p^(i+1)) as p runs through the primes.

Cf. A001248, A131991, A131992, A131993, A253905.

Cf. A008864, A000203. - Zak Seidov, Feb 13 2016

[p^2+p+1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 20 2014

A060800:= n -> map (p -> p^(2)+p+1, ithprime(n)):
seq (A060800(n), n=1..41); # Jani Melik, Jan 25 2011

#^2 + # + 1&/@Prime[Range[200]] (* Vincenzo Librandi, Mar 20 2014 *)

{ n=0; forprime (p=2, prime(1000), write("b060800.txt", n++, " ", p^2 + p + 1); ) } \\ Harry J. Smith, Jul 13 2009

a(n) = A036690(n) + 1.
a(n) = 1 + A008864(n)*A000040(n) = (A030078(n) - 1)/A006093(n). - Reinhard Zumkeller, Aug 06 2007
a(n) = sigma(prime(n)^2) = A000203(A000040(n)^2). - Zak Seidov, Feb 13 2016
a(n) = A000203(A001248(n)). - Michel Marcus, Feb 15 2016
Product_{n>=1} (1 - 1/a(n)) = zeta(3)/zeta(2) (A253905). - Amiram Eldar, Nov 07 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001

A059269 Numbers m for which the number of divisors, tau(m), is divisible by 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 24 2001

tau(n) is divisible by 3 iff at least one prime in the prime factorization of n has exponent of the form 3*m + 2. This sequence is an extension of the sequence A038109 in which the numbers has at least one prime with exponent 2 (the case of m = 0 here ) in their prime factorization.
The union of A211337 and A211338 is the complementary sequence to this one. - Douglas Latimer, Apr 12 2012
Numbers whose cubefree part (A050985) is not squarefree (A005117). - Amiram Eldar, Mar 09 2021

			a(7) = 28 is a term because the number of divisors of 28, d(28) = 6, is divisible by 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
S. S. Pillai, On a congruence property of the divisor function, J. Indian Math. Soc. (N. S.), Vol. 6, (1942), pp. 118-119.
L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Cf. A000005, A005117, A038109, A050985, A211337, A211338, A253905.

Characteristic function: A353470.

with(numtheory): for n from 1 to 1000 do if tau(n) mod 3 = 0 then printf(`%d,`,n) fi: od:

Select[Range[230], Divisible[DivisorSigma[0, #], 3] &] (* Amiram Eldar, Jul 26 2020 *)

is(n)=vecmax(factor(n)[,2]%3)==2 \\ Charles R Greathouse IV, Apr 10 2012

is(n)=numdiv(n)%3==0 \\ Charles R Greathouse IV, Sep 18 2015

Conjecture: a(n) ~ k*n where k = 1/(1 - Product(1 - (p-1)/(p^(3*i)))) = 3.743455... where p ranges over the primes and i ranges over the positive integers. - Charles R Greathouse IV, Apr 13 2012
The asymptotic density of this sequence is 1 - zeta(3)/zeta(2) = 1 - 6*zeta(3)/Pi^2 = 0.2692370305... (Sathe, 1945). Therefore, the above conjecture, a(n) ~ k*n, is true, but k = 1/(1-6*zeta(3)/Pi^2) = 3.7141993349... - Amiram Eldar, Jul 26 2020
A001248 UNION A030515 UNION A030627 UNION A030630 UNION A030633 UNION A030636 UNION ... - R. J. Mathar, May 05 2023

More terms from James Sellers, Jan 24 2001

A072905 a(n) is the least k > n such that k*n is a square.

Original entry on oeis.org

4, 8, 12, 9, 20, 24, 28, 18, 16, 40, 44, 27, 52, 56, 60, 25, 68, 32, 76, 45, 84, 88, 92, 54, 36, 104, 48, 63, 116, 120, 124, 50, 132, 136, 140, 49, 148, 152, 156, 90, 164, 168, 172, 99, 80, 184, 188, 75, 64, 72, 204, 117, 212, 96, 220, 126, 228, 232, 236, 135, 244, 248

Offset: 1

Benoit Cloitre, Aug 10 2002

From Peter Kagey, Jun 22 2015: (Start)
a(n) is a bijection from the positive integers to A013929 (numbers that are not squarefree). Proof:
(1) Injection: Suppose that b(2) Surjection: Given some number k in A013929, a(A007913(k)*(A000188(k)-1)^2.) = k (End)

			12 is the smallest integer > 3 such that 3*12 = 6^2 is a perfect square, hence a(3) = 12.

Peter Kagey, Table of n, a(n) for n = 1..5000

Cf. A000188, A007913, A010052, A067722.

Cf. A182448, A253905.

a072905 n = head [k | k <- [n + 1 ..], a010052 (k * n) == 1]
-- Reinhard Zumkeller, Feb 07 2015

f:= proc(n) local F,f,x,y;
     F:= ifactors(n)[2];
     x:= mul(`if`(f[2]::odd,f[1],1),f=F);
     y:= mul(f[1]^floor(f[2]/2),f=F);
     x*(y+1)^2
end proc:
map(f, [$1..100]); # Robert Israel, Jun 23 2015

a[n_] := For[k = n+1, True, k++, If[IntegerQ[Sqrt[k*n]], Return[k]]]; Array[a, 100] (* Jean-François Alcover, Jan 26 2018 *)

a(n)=if(n<0,0,s=n+1; while(issquare(s*n)==0,s++); s)

a(n)=my(c=core(n)); (sqrtint(n/c)+1)^2*c \\ Charles R Greathouse IV, Jun 23 2015

def a(n)
  k = Math.sqrt(n).to_i
  k -= 1 until n % k**2 == 0
  n + 2*n/k + n/(k**2)
end # Peter Kagey, Jul 27 2015

a(n) = n + A067722(n). - Peter Kagey, Feb 05 2015
a(n) = A007913(n)*(A000188(n)+1)^2. - Peter Kagey, Feb 06 2015
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + 2*zeta(3)/zeta(2) + Pi^2/15 = 3.11949956554216757204... . - Amiram Eldar, Feb 17 2024

A029939 a(n) = Sum_{d|n} phi(d)^2.

Original entry on oeis.org

1, 2, 5, 6, 17, 10, 37, 22, 41, 34, 101, 30, 145, 74, 85, 86, 257, 82, 325, 102, 185, 202, 485, 110, 417, 290, 365, 222, 785, 170, 901, 342, 505, 514, 629, 246, 1297, 650, 725, 374, 1601, 370, 1765, 606, 697, 970, 2117, 430, 1801, 834, 1285, 870, 2705, 730, 1717, 814, 1625

Offset: 1

N. J. A. Sloane

Equals the inverse Mobius transform (A051731) of A127473. - Gary W. Adamson, Aug 20 2008

Number of (i,j) in {1,2,...,n}^2 such that gcd(n,i) = gcd(n,j). - Benoit Cloitre, Dec 31 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051731, A062367, A065463, A127473, A253905.

with(numtheory): A029939 := proc(n) local i,j; j := 0; for i in divisors(n) do j := j+phi(i)^2; od; j; end;
# alternative
N:= 1000: # to get a(1)..a(N)
A:= Vector(N,1):
for d from 2 to N do
  pd:= numtheory:-phi(d)^2;
  md:= [seq(i,i=d..N,d)];
  A[md]:= map(`+`,A[md],pd);
od:
seq(A[i],i=1..N); # Robert Israel, May 30 2016

Table[Total[EulerPhi[Divisors[n]]^2],{n,60}] (* Harvey P. Dale, Feb 04 2017 *)
f[p_, e_] := (p^(2*e)*(p-1)+2)/(p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = sumdiv(n, d, eulerphi(d)^2); \\ Michel Marcus, Jan 17 2017

Multiplicative with a(p^e) = (p^(2*e)*(p-1)+2)/(p+1). - Vladeta Jovovic, Nov 19 2001
G.f.: Sum_{k>=1} phi(k)^2*x^k/(1 - x^k), where phi(k) is the Euler totient function (A000010). - Ilya Gutkovskiy, Jan 16 2017
a(n) = Sum_{k=1..n} phi(n/gcd(n, k)). - Ridouane Oudra, Nov 28 2019
Sum_{k>=1} 1/a(k) = 2.3943802654751092440350752246012273573942903149891228695146514601814537713... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/(3*zeta(2))) * Product_{p prime} (1 - 1/(p*(p+1))) = A253905 * A065463 / 3 = 0.171593... . - Amiram Eldar, Oct 25 2022

A351265 Sum of the squares of the squarefree divisors of n.

Original entry on oeis.org

1, 5, 10, 5, 26, 50, 50, 5, 10, 130, 122, 50, 170, 250, 260, 5, 290, 50, 362, 130, 500, 610, 530, 50, 26, 850, 10, 250, 842, 1300, 962, 5, 1220, 1450, 1300, 50, 1370, 1810, 1700, 130, 1682, 2500, 1850, 610, 260, 2650, 2210, 50, 50, 130, 2900, 850, 2810, 50, 3172, 250, 3620

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Inverse Möbius transform of n^2 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

			a(6) = 50; a(6) = Sum_{d|6} d^2 * mu(d)^2 = 1^2*1 + 2^2*1 + 3^2*1 + 6^2*1 = 50.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A008683 (mu), A253905, A328639, A322360.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), this sequence (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^2); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n],SquareFreeQ]^2],{n,80}] (* Harvey P. Dale, Dec 26 2024 *)

a(n) = sumdiv(n, d, if (issquarefree(d), d^2)); \\ Michel Marcus, Feb 06 2022

a(n) = Sum_{d|n} d^2 * mu(d)^2.
a(n) = abs(A328639(n)).
G.f.: Sum_{k>=1} mu(k)^2 * k^2 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^2. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(3*zeta(2)) = A253905 / 3 = 0.243587... . - Amiram Eldar, Nov 10 2022
Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s-4). - Michael Shamos, Aug 05 2023

A368712 The maximal exponent in the prime factorization of the cubefree numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jan 04 2024

The asymptotic density of occurrences of 1 is zeta(3)/zeta(2) = 0.730762... (A253905), and the asymptotic density of occurrences of 2 is 1 - zeta(3)/zeta(2) = 0.269237... .

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

Cf. A004709, A005117, A008966, A013929, A033150, A051903, A067259.
Cf. A002117, A013661, A253905.
Similar sequences: A368710, A368711, A368713.

s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]
(* or *)
f[n_] := Module[{e = Max @@ FactorInteger[n][[;; , 2]]}, If[e < 3, e, Nothing]]; f[1] = 0; Array[f, 100]

lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(e < 3, print1(e, ", ")));}

from sympy import mobius, integer_nthroot, factorint
def A368712(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return max(factorint(m).values(),default=0) # Chai Wah Wu, Aug 12 2024

a(n) = A051903(A004709(n)).
a(n) = 2 - A008966(A004709(n)) for n >= 2.
Except for n = 1, a(n) = 1 or 2.
a(n) = 1 if and only if A004709(n) is squarefree (A005117).
a(n) = 2 if and only if A004709(n) > 1 and is nonsquarefree (A013929), i.e., A004709(n) is in A067259.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - zeta(3)/zeta(2) = 2 - A253905 = 1.269237030598... .

A371188 Indices of the squarefree numbers in the sequence of cubefree numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 40, 41, 44, 46, 47, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 79, 80, 81, 82, 86, 87, 88, 89, 90, 91

Offset: 1

Amiram Eldar, Mar 14 2024

The asymptotic density of this sequence is zeta(3)/zeta(2) = 0.730762... (A253905).

			The first 5 cubefree numbers are 1, 2, 3, 4, and 5. The 1st, 2nd, 3rd, and 5th, 1, 2, 3, and 5, are squarefree. Therefore, the first 4 terms of this sequence are 1, 2, 3, and 5.

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

Cf. A004709, A005117.

Cf. A253905, A306633.

freeQ[n_, k_] := AllTrue[FactorInteger[n], Last[#] < k &]; Position[Select[Range[200], freeQ[#, 3] &], _?(freeQ[#1, 2] &), Heads -> False] // Flatten

isfree(n, k) = n == 1 || vecmax(factor(n)[, 2]) < k;
lista(kmax) = {my(c = 0); for(k = 1, kmax, if(isfree(k, 3), c++; if(isfree(k, 2), print1(c, ", "))));}

A004709(a(n)) = A005117(n).

a(n) ~ c * n, where c = zeta(2)/zeta(3) = 1.368432... (A306633).

A336590 Numbers k such that k/A008834(k) is squarefree, where A008834(k) is the largest cube dividing k.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

Numbers such that none of the exponents in their prime factorization is of the form 3*m + 2.
Cohen (1962) proved that for a given number k >= 2 the asymptotic density of numbers whose exponents in their prime factorization are of the forms k*m or k*m + 1 only is zeta(k)/zeta(2). In this sequence k = 3, and therefore its asymptotic density is zeta(3)/zeta(2) = 6*zeta(3)/Pi^2 = 0.7307629694... (A253905).
Also, numbers k whose number of divisors, A000005(k), is not divisible by 3, i.e., complement of A059269.

			6 is a term since 6 = 2^1 * 3^1 and 1 is not of the form 3*m + 2.
9 is not a term since 9 = 3^2 and 2 is of the form 3*m + 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes. III. Certain equally distributed sets of integers, Pacific Journal of Mathematics, No. 12, Vol. 1 (1962), pp. 77-84.
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
L. G. Sathe, On a congruence property of the divisor function, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.

Cf. A000005, A002117, A005117, A008834, A013661, A059269, A253905.

Union of A211337 and A211338.

Select[Range[100], Max[Mod[FactorInteger[#][[;;,2]], 3]] < 2 &]

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A023900 Dirichlet inverse of Euler totient function (A000010).

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A306633 Decimal expansion of zeta(2)/zeta(3).

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A060800 a(n) = p^2 + p + 1 where p runs through the primes.

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A059269 Numbers m for which the number of divisors, tau(m), is divisible by 3.

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A072905 a(n) is the least k > n such that k*n is a square.

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A029939 a(n) = Sum_{d|n} phi(d)^2.