A064549 -id:A064549 - cp's OEIS frontend

A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

1, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 2, 4, 2, 1, 4, 1, 2, 1, 8, 1, 16, 1, 2, 2, 2, 1, 4, 1, 4, 4, 2, 1, 4, 1, 2, 2, 2, 1, 16, 1, 2, 1, 8, 4, 4, 1, 2, 8, 4, 1, 4, 1, 2, 2, 2, 1, 16, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 16, 1, 4, 2, 2, 1, 128, 1, 2, 2, 4, 1, 4, 1, 2, 8, 4, 1, 4, 1, 4, 1, 2, 4, 16, 4, 2, 2, 2, 1, 8

Offset: 1

Antti Karttunen, Aug 30 2018

These are also denominators (in their lowest terms) for the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.
From Antti Karttunen, Sep 02 2018: (Start)
Proof for the above claim:
This sequence is defined as the denominator (given in the lowest terms) of rational valued function r(1) = 1, r(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. Define sequence Ay(n) as the denominator of function s(n), with otherwise similar definition, but with A064549 in place of A000290. Let Ay(n) be the denominator of s(n), reduced also into the lowest terms. (Corresponding numerators are A318649 and A318511 respectively. Note that the denominators in both cases must always be of the form 2^k, with k >= 0).
  By applying the distributive property of Dirichlet Convolution [which says that for any completely multiplicative function f, it doesn't matter whether one multiplies the result of convolution afterwards, or whether one multiplies the operands separately before convolution: f(g * g) = (fg) * (fg)], with A000027 in the role of f in both cases, one obtains a pair of equations:
    A318649(n)     A318681(n)     n*A299149(n)
    ----------  =  ----------  =  ------------
    A318512(n)     A299150(n)      A299150(n)
  and
    A318511(n)     A318680(n)     n*A318653(n)
    ----------  =  ----------  =  ------------
       Ay(n)       A299150(n)      A299150(n)
  where the leftmost ratios are reduced into their lowest terms.
  Sequence A318656 gives the 2-adic valuation of ratio A318649(n)/A318512(n), and because there are no even terms neither in A299149 nor in A318653, it also gives the 2-adic valuation of the latter ratio. As A318511/Ay is given in the lowest terms (not both of A318511(n) and Ay(n) can be even at same n), this implies that Ay must indeed be identical to A318512, and furthermore that A318655(n) = A007814(A318649(n)) = A007814(A318511(n)).
(End)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Wikipedia, Dirichlet convolution

Cf. A000290, A064549, A046644, A299150, A318513, A318651, A318653, A318654, A318655, A318656, A318658, A318680, A318681.

Cf. A318649, A318511 (numerators).

f[1] = 1; f[n_] := f[n] = 1/2 (n*Times @@ FactorInteger[n][[All, 1]] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Denominator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 10 2025 *)

up_to = 65537;
A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); };
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&&dA064549(n)));
A318512(n) = denominator(v318511_12[n]);

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p^2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. [Equally, one could use A064549 in place of A000290.]
a(n) = 2^A318513(n).
a(n) = A046644(n)/A318651(n).
a(2n-1) = A046644(2n-1) = A318658(2n-1), for all n >= 1.

The main definition changed, more formulas added by Antti Karttunen, Aug 31 2018

A318511 Numerators of the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.

Original entry on oeis.org

1, 2, 9, 2, 25, 9, 49, 4, 27, 25, 121, 9, 169, 49, 225, 6, 289, 27, 361, 25, 441, 121, 529, 18, -125, 169, 405, 49, 841, 225, 961, 12, 1089, 289, 1225, 27, 1369, 361, 1521, 50, 1681, 441, 1849, 121, 675, 529, 2209, 27, -1029, -125, 2601, 169, 2809, 405, 3025, 98, 3249, 841, 3481, 225, 3721, 961, 1323, 20, 4225, 1089, 4489, 289, 4761, 1225, 5041, 27

Offset: 1

Antti Karttunen, Aug 30 2018

No zeros among the first 2^20 terms.

For odd primes p, it seems that a(p) = p^2.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A064549, A318512 (denominators).

Cf. also A317935.

f[1] = 1; f[n_] := f[n] = 1/2 (n*Times @@ FactorInteger[n][[All, 1]] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 10 2025 *)

up_to = 65537;
A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); };
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&&dA064549(n)));
A318511(n) = numerator(v318511_12[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064549(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 10 2025: (Start)
Let f(s) = Product_{p prime} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
Sum_{k=1..n} A318511(k) / A318512(k) ~ n^3 * sqrt(Pi*f(3)/(54*log(n))) * (1 + (1/3 - gamma/2 - f'(3)/(2*f(3)) - 3*zeta'(2)/Pi^2) / (2*log(n))), where
f(3) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144...
f'(3)/f(3) = Sum_{p prime} (3*p - 2) * log(p) / (p^3 - 2*p + 1) = 1.469536740824614833203393993450164364663334798759143895712...
and gamma is the Euler-Mascheroni constant A001620. (End)

A363171 Numbers k such that A064549(k) is an abundant number (A005101).

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 28, 30, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186

Offset: 1

Amiram Eldar, May 19 2023

nonn

First differs from A334166 at n = 21.
The least odd term is a(33) = 105, and the least term that is coprime to 6 is a(11850456) = 37182145.
The ordered values of A003557(A363169(n)): m is a powerful abundant number (A363169) if and only if A003557(m) is in this sequence.
If k is a term then any positive multiple of k is also a term. The primitive terms are in A363172.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 2, 30, 322, 3201, 31863, 318336, 3188014, 31855257, 318427893, 3184885813, 31853300276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3185... .

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

Cf. A003557, A064549, A334166.

Subsequences: A005101, A363169, A363172.

q[n_] := DivisorSigma[-1, n * Times @@ FactorInteger[n][[;; , 1]]] > 2; Select[Range[200], q]

A064549(n) = { my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]+1)); };
is(n) = sigma(A064549(n), -1) > 2;

A348018 a(n) is the index of A064549(n) = n * Product_{p prime|n} p in the sequence of powerful numbers (A001694).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 10, 5, 7, 14, 16, 12, 20, 21, 24, 8, 28, 15, 31, 22, 35, 36, 39, 19, 17, 44, 13, 32, 48, 50, 51, 11, 56, 59, 60, 23, 65, 67, 68, 33, 71, 73, 75, 52, 43, 81, 84, 27, 30, 37, 90, 64, 94, 29, 97, 46, 102, 104, 107, 74, 110, 111, 62, 18, 117, 119

Offset: 1

Amiram Eldar, Sep 24 2021

nonn

A permutation of the positive integers.

The inverse permutation of A306458.

			The sequence of powerful numbers (A001694) begins with 1, 4, 8, 9, ...
The position of A064549(1) = 1 in A001694 is 1, so a(1) = 1.
The position of A064549(3) = 9 in A001694 is 4, so a(3) = 4.

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

Cf. A001694, A064549, A306458.

powQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; pows = Join[{1}, Select[Range[10^4], powQ]]; TakeWhile[Table[FirstPosition[pows, n * Times @@ (First /@ FactorInteger[n])][[1]], {n, 1, 100}], NumericQ]

A001694(a(n)) = A064549(n).
A306458(a(n)) = a(A306458(n)) = n.
The fixed points of this permutation are 1, 2, 12, 1208, 1256, 1288 and no more below 3*10^5.

A003557 n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7

Offset: 1

Marc LeBrun

a(n) is the size of the Frattini subgroup of the cyclic group C_n - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001.
Also of the Frattini subgroup of the dihedral group with 2*n elements. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002
Number of solutions to x^m==0 (mod n) provided that n < 2^(m+1), i.e. the sequence of sequences A000188, A000189, A000190, etc. converges to this sequence. - Henry Bottomley, Sep 18 2001
a(n) is the number of nilpotent elements in the ring Z/nZ. - Laszlo Toth, May 22 2009
The sequence of partial products of a(n) is A085056(n). - Peter Luschny, Jun 29 2009
The first occurrence of n in this sequence is at A064549(n). - Franklin T. Adams-Watters, Jul 25 2014
From Hal M. Switkay, Jul 03 2025: (Start)
For n > 1, a(n) is a proper divisor of n. Thus the sequence n, a(n), a(a(n)), ... eventually becomes 1. This yields a minimal factorization of n as a product of squarefree numbers (A005117), each factor dividing all larger factors, in a factorization that is conjugate to the minimal factorization of n as a product of prime powers (A000961), as follows.
Let f(n,0) = n, and let f(n,k) = a(f(n,k-1)) for k > 0. A051903(n) is the minimal value of k such that f(n,k) = 1. A051903(n) <= log(n)/log(2). Since n/a(n) = A007947(n) is always squarefree by definition, n is a product of squarefree factors in the form Product_{i=1..A051903(n)} [f(n,i-1)/f(n,i)].
The two factorizations correspond to conjugate partitions of bigomega(n) = A001222(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Steven R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Cf. A007947, A062378, A062379, A064549, A300717 (Möbius transform), A326306 (inv. Möbius transf.), A328572.
Sequences that are multiples of this sequence (the other factor of a pointwise product is given in parentheses): A000010 (A173557), A000027 (A007947), A001615 (A048250), A003415 (A342001), A007434 (A345052), A057521 (A071773).
Cf. A082695 (Dgf at s=2), A065487 (Dgf at s=3).
Cf. A000961, A001222, A005117, A051903.

a003557 n = product $ zipWith (^)
                      (a027748_row n) (map (subtract 1) $ a124010_row n)
-- Reinhard Zumkeller, Dec 20 2013

using Nemo
function A003557(n)
    n < 4 && return 1
    q = prod([p for (p, e) ∈ Nemo.factor(fmpz(n))])
    return n == q ? 1 : div(n, q)
end
[A003557(n) for n in 1:90] |> println  # Peter Luschny, Feb 07 2021

[(&+[(Floor(k^n/n)-Floor((k^n-1)/n)): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Nov 02 2018

A003557 := n -> n/ilcm(op(numtheory[factorset](n))):
seq(A003557(n), n=1..98); # Peter Luschny, Mar 23 2011
seq(n / NumberTheory:-Radical(n), n = 1..98); # Peter Luschny, Jul 20 2021

Prepend[ Array[ #/Times@@(First[ Transpose[ FactorInteger[ # ] ] ])&, 100, 2 ], 1 ] (* Olivier Gérard, Apr 10 1997 *)

a(n)=n/factorback(factor(n)[,1]) \\ Charles R Greathouse IV, Nov 17 2014

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 20 2020

from sympy.ntheory.factor_ import core
from sympy import divisors
def a(n): return n / max(i for i in divisors(n) if core(i) == i)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 16 2017

from math import prod
from sympy import primefactors
def A003557(n): return n//prod(primefactors(n)) # Chai Wah Wu, Nov 04 2022

def A003557(n) : return n*mul(1/p for p in prime_divisors(n))
[A003557(n) for n in (1..98)] # Peter Luschny, Jun 10 2012

Multiplicative with a(p^e) = p^(e-1). - Vladeta Jovovic, Jul 23 2001
a(n) = n/rad(n) = n/A007947(n) = sqrt(J_2(n)/J_2(rad(n))), where J_2(n) is A007434. - Enrique Pérez Herrero, Aug 31 2010
a(n) = (J_k(n)/J_k(rad(n)))^(1/k), where J_k is the k-th Jordan Totient Function: (J_2 is A007434 and J_3 A059376). - Enrique Pérez Herrero, Sep 03 2010
Dirichlet convolution of A000027 and A097945. - R. J. Mathar, Dec 20 2011
a(n) = A000010(n)/|A023900(n)|. - Eric Desbiaux, Nov 15 2013
a(n) = Product_{k = 1..A001221(n)} (A027748(n,k)^(A124010(n,k)-1)). - Reinhard Zumkeller, Dec 20 2013
a(n) = Sum_{k=1..n}(floor(k^n/n)-floor((k^n-1)/n)). - Anthony Browne, May 11 2016
a(n) = e^[Sum_{k=2..n} (floor(n/k)-floor((n-1)/k))*(1-A010051(k))*Mangoldt(k)] where Mangoldt is the Mangoldt function. - Anthony Browne, Jun 16 2016
a(n) = Sum_{d|n} mu(d) * phi(d) * (n/d), where mu(d) is the Moebius function and phi(d) is the Euler totient function (rephrases formula of Dec 2011). - Daniel Suteu, Jun 19 2018
G.f.: Sum_{k>=1} mu(k)*phi(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
Dirichlet g.f.: Product_{primes p} (1 + 1/(p^s - p)). - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k).
a(n) = Sum_{k=1..n} mu(gcd(n,k))*(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = A001615(n)/A048250(n) = A003415/A342001(n) = A057521(n)/A071773(n). - Antti Karttunen, Jun 08 2021

Secondary definition added to the name by Antti Karttunen, Jun 08 2021

A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=bc^2d^3 then a(n)=c^2*d^3 when b is minimized.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 8, 1, 9, 1, 4, 1, 1, 1, 32, 1

Offset: 1

Henry Bottomley, Sep 01 2000

			a(40) = 8 since 40 = 2^3 * 5 so the powerful part is 2^3 = 8.

Antti Karttunen, Table of n, a(n) for n = 1..16383 (first 1000 terms from T. D. Noe)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000000 terms).
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences.
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seq., Vol. 6 (2003), Article 03.3.4.
Index entries for sequences related to powerful numbers.

Cf. A001694, A003415, A003557, A055231, A064549.

A057521 := proc(n)
    local a,d,e,p;
    a := 1;
    for d in ifactors(n)[2] do
        e := d[1] ;
        p := d[2] ;
        if e > 1 then
            a := a*p^e ;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Jun 09 2016

rad[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := n/Denominator[n/rad[n]^2]; Table[a[n], {n, 1, 97}] (* Jean-François Alcover, Jun 20 2013 *)
f[p_, e_] := If[e > 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

a(n)=my(f=factor(n));prod(i=1,#f~,if(f[i,2]>1,f[i,1]^f[i,2],1)) \\ Charles R Greathouse IV, Aug 13 2013

a(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,2]==1, f[i,1]=1)); factorback(f); \\ Michel Marcus, Jan 29 2021

from sympy import factorint, prod
def a(n): return 1 if n==1 else prod(1 if e==1 else p**e for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

from math import prod
from sympy import factorint
def A057521(n): return n//prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022

a(n) = n / A055231(n).
Multiplicative with a(p)=1 and a(p^e)=p^e for e>1. - Vladeta Jovovic, Nov 01 2001
From Antti Karttunen, Nov 22 2017: (Start)
a(n) = A064549(A003557(n)).
A003557(a(n)) = A003557(n).
(End)
a(n) = gcd(n, A003415(n)^k), for all k >= 2. [This formula was found in the form k=3 by Christian Krause's LODA miner. See Ufnarovski and Åhlander paper, Theorem 5 on p. 4 for why this holds] - Antti Karttunen, Mar 09 2021
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/ p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1)). - Amiram Eldar, Sep 18 2023
From Vaclav Kotesovec, Apr 09 2025, simplified May 11 2025: (Start)
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 - 1/p^(3*s-2) + 1/p^(3*s-3) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 2/p^(3/2) - 1/p^(5/2)) = 3.51955505841710664719752940369857817... = A328013. (End)

A092261 Sum of unitary, squarefree divisors of n, including 1.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 1, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 4, 1, 42, 1, 8, 30, 72, 32, 1, 48, 54, 48, 1, 38, 60, 56, 6, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 3, 72, 8, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 1, 74, 114, 4, 20, 96, 168, 80

Offset: 1

Steven Finch, Feb 20 2004

Unitary convolution of the sequence of n*mu^2(n) (absolute values of A055615) and A000012. - R. J. Mathar, May 30 2011

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr. 74 (1960) 66-80, sequence sigma'(n).
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

Cf. A000203, A003557, A007947, A048250, A055231, A056671, A057521, A065465, A295294, A295295, A329728.

Table[Plus @@ Select[Divisors@ n, Max @@ Last /@ FactorInteger@ # == 1 && GCD[#, n/#] == 1 &], {n, 1, 79}] (* Michael De Vlieger, Mar 08 2015 *)
f[p_, e_] := If[e==1, p+1, 1]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 79] (* Amiram Eldar, Mar 01 2019 *)

a(n) = sumdiv(n, d, d*issquarefree(d)*(gcd(d, n/d) == 1)); \\ Michel Marcus, Mar 06 2015

for(n=1, 100, print1(direuler(p=2, n, (1 + p^2*X^3 - p*X^2 - p^2*X^2)/(1-X)/(1-p*X))[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

;; This implementation utilizes the memoization-macro definec for which an implementation is available at http://oeis.org/wiki/Memoization#Scheme
;; The other functions, A020639, A067029 and A028234 can be found under the respective entries, and should likewise defined with definec:
(definec (A092261 n) (if (= 1 n) 1 (* (+ 1 (if (> (A067029 n) 1) 0 (A020639 n))) (A092261 (A028234 n))))) ;; Antti Karttunen, Nov 25 2017

Multiplicative with a(p) = p+1 and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Feb 22 2004
From Álvar Ibeas, Mar 06 2015: (Start)
a(n) = a(A055231(n)) = A000203(A055231(n)).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(1-2s)).
(End)
From Antti Karttunen, Nov 25 2017: (Start)
a(n) = A048250(A055231(n)).
a(n) = A000203(n) / A295294(n).
a(n) = A048250(n) / A295295(n) = A048250(n) / A048250(A057521(n)), where A057521(n) = A064549(A003557(n)).
(End)
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - Amiram Eldar, Jun 10 2020
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(2-3*s) - p^(1-2*s) - p^(2-2*s)). - Vaclav Kotesovec, Aug 20 2021
a(n) = Sum_{d|n, gcd(d,n/d)=1} d * mu(d)^2. - Wesley Ivan Hurt, May 26 2023

A078310 a(n) = n*rad(n) + 1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 10, 9, 26, 37, 50, 17, 28, 101, 122, 73, 170, 197, 226, 33, 290, 109, 362, 201, 442, 485, 530, 145, 126, 677, 82, 393, 842, 901, 962, 65, 1090, 1157, 1226, 217, 1370, 1445, 1522, 401, 1682, 1765, 1850, 969, 676, 2117, 2210, 289, 344, 501, 2602, 1353

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

A112526(a(n) - 1) = 1, see also A224866. - Reinhard Zumkeller, Jul 23 2013

Increase each exponent in the prime factorization by one, then add 1 to the new product. - M. F. Hasler, Jan 22 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Smallest, greatest factor: A078311, A078312, number of factors: A078313, A078314, min, max exponent: A078315, A078316, number, sum of divisors: A078317, A078318, sum of prime factors: A078319, A078320, Euler's totient: A078321, squarefree kernel: A078322, arithmetic derivative: A078323.

Cf. primes: A078324, squarefree: A078325, squareful: A078326.

a078310 n = n * a007947 n + 1
-- Reinhard Zumkeller, Jul 23 2013, Oct 19 2011

a:= n-> 1+n*mul(i[1], i=ifactors(n)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 22 2017

A078310[n_] := n*Times @@ FactorInteger[n][[All, 1]] + 1; Array[A078310, 50] (* G. C. Greubel, Apr 25 2017 *)

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
a(n)=n*rad(n)+1 \\ Charles R Greathouse IV, Jul 09 2013

a(n)={n=factor(n);n[,2]+=vectorv(matsize(n)[1],i,1);factorback(n)+1} \\ M. F. Hasler, Jan 22 2017

a(n)=prod(k=1,matsize(n=factor(n))[1],n[k,1]^(n[k,2]+1))+1 \\ M. F. Hasler, Jan 22 2017

a(n) = A064549(n)+1.

A280292 a(n) = sopfr(n) - sopf(n).

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Dec 31 2016

nonn

Alladi and Erdős (1977) proved that for all numbers m>=0, m!=1, the sequence of numbers k such that a(k) = m has a positive asymptotic density which is equal to a rational multiple of 1/zeta(2) = 6/Pi^2 (A059956). For example, when m=0, the sequence is the squarefree numbers (A005117), whose density is 6/Pi^2, and when m=2 the sequence is A081770, whose density is 1/Pi^2. - Amiram Eldar, Nov 02 2020

Sum of prime factors minus sum of distinct prime factors. Counting partitions by this statistic (sum minus sum of distinct parts) gives A364916. - Gus Wiseman, Feb 21 2025

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See pp. 164-166.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 165.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294, alternative link.
Jean-Marie De Koninck, Paul Erdős and Aleksandar Ivić, Reciprocals of certain large additive functions, Canadian Mathematical Bulletin, Vol. 24, No. 2 (1981), pp. 225-231.
Aleksandar Ivić, On certain large additive functions, arXiv:math/0311505 [math.NT], 2003.

Cf. A001414 (sopfr), A008472 (sopf), A057521, A059956, A081770, A280163, A338559, A338560.
A multiplicative version is A003557, firsts A064549 (sorted A001694).
For length instead of sum we have A046660.
For product instead of sum we have A066503, firsts A381076.
Positions of first appearances are A280286 (sorted A381075).
For indices instead of factors we have A380955, firsts A380956 (sorted A380957).
For exponents instead of factors we have A380958, firsts A380989.
A000040 lists the primes, differences A001223.
A001222 counts prime factors (distinct A001221).
A003963 gives product of prime indices, distinct A156061, excess A380986.
A005117 lists squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
A027746 lists prime factors, distinct A027748.
A112798 lists prime indices (sum A056239), distinct A304038 (sum A066328).
Cf. A071625, A075255, A116861, A136565, A175508, A290106, A364916, A381077.

Array[Total@ # - Total@ Union@ # &@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 105] (* Michael De Vlieger, Feb 25 2019 *)

sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j, 1]);
a(n) = sopfr(n) - sopf(n);

a(n) = A001414(n) - A008472(n).
a(A005117(n)) = 0.
a(n) = A001414(A003557(n)). - Antti Karttunen, Oct 07 2017
Additive with a(1) = 0 and a(p^e) = p*(e-1) for prime p and e > 0. - Werner Schulte, Feb 24 2019
From Amiram Eldar, Nov 02 2020: (Start)
a(n) = a(A057521(n)).
Sum_{n<=x} a(n) ~ x*log(log(x)) + O(x) (Alladi and Erdős, 1977).
Sum_{n<=x, n nonsquarefree} 1/a(n) ~ c*x + O(sqrt(x)*log(x)), where c = Integral_{t=0..1} (F(t)-6/Pi^2)/t dt, and F(t) = Product_{p prime} (1-1/p)*(1-1/(t^p - p)) (De Koninck et al., 1981; Finch, 2018), or, equivalently c = Sum_{k>=2} d(k)/k = 0.1039..., where d(k) = (6/Pi^2)*A338559(k)/A338560(k) is the asymptotic density of the numbers m with a(m) = k (Alladi and Erdős, 1977; Ivić, 2003). (End)

More terms from Antti Karttunen, Oct 07 2017

A291750 Compound filter: a(n) = P(A003557(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 4, 7, 8, 16, 67, 29, 19, 18, 154, 67, 80, 92, 277, 277, 53, 154, 94, 191, 173, 497, 631, 277, 109, 50, 862, 75, 302, 436, 2557, 497, 169, 1129, 1432, 1129, 142, 704, 1771, 1541, 214, 862, 4561, 947, 668, 328, 2557, 1129, 179, 98, 236, 2557, 905, 1432, 199, 2557, 355, 3161, 4006, 1771, 2630, 1892, 4561, 564, 593, 3487, 10297, 2279, 1487, 4561, 10297, 2557

Offset: 1

Antti Karttunen, Sep 04 2017

nonn

A000203 (sigma(n)) is a function of this sequence, because formula
  A000203(n) = A092261(n) * A295294(n)
can be rewritten as a relation:
  A000203(n) = A000203(A057521(n)) * A048250(n) / A048250(A057521(n)),
where A057521(n) = A064549(A003557(n)), thus A000203(n) is a function of A003557(n) and A048250(n), the values that are packed here into a(n).
A001615 (Dedekind's psi) is a function of this sequence, because it can be written as A001615(n) = A003557(n)*A048250(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000027, A000203, A001615, A003557, A048250, A291751 (rgs-version of this filter).

A003557(n) = n/factorback(factor(n)[, 1]); \\ This function from Charles R Greathouse IV, Nov 17 2014
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));

a(n) = (1/2)*(2 + ((A003557(n) + A048250(n))^2) - A003557(n) - 3*A048250(n)).

cp's OEIS Frontend

A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

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A318511 Numerators of the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.

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A363171 Numbers k such that A064549(k) is an abundant number (A005101).

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A348018 a(n) is the index of A064549(n) = n * Product_{p prime|n} p in the sequence of powerful numbers (A001694).

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A003557 n divided by largest squarefree divisor of n; if n = Product p(k)^e(k) then a(n) = Product p(k)^(e(k)-1), with a(1) = 1.

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A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=b*c^2*d^3 then a(n)=c^2*d^3 when b is minimized.

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A057521 Powerful (1) part of n: if n = Product_i (pi^ei) then a(n) = Product_{i : ei > 1} (pi^ei); if n=bc^2d^3 then a(n)=c^2*d^3 when b is minimized.